Quadratic Equation: Formulas, Solutions, Graphs, and Examples

Are you looking to demystify the world of quadratic equations? Whether you’re a student, educator, or simply curious about mathematics, this guide offers a thorough exploration of quadratic equations, including their definition, quadratic formula, graphical representations, and practical applications. Dive in to discover how these equations are integral to various fields and how you can solve them with ease.

Part 1. What is the Quadratic Equation?

Definition of Quadratic Equation

The quadratic equation is central to algebra, and it arises in many forms throughout mathematics and its applications. A quadratic equation is defined by the largest power of the variable, which is squared. The conventional form is ax^2 +bx +c =0

where ‘a’ does not equal 0. The roots of this equation represent the values of ‘x’ that satisfy the equation. It’s noteworthy to note that a quadratic equation always has two roots, which can be real or complex, depending on the discriminant.

In our daily lives, quadratic equations are more present than we might think. From calculating the area of rooms to determining the profit of a product in business, or even finding the speed of an object in motion, quadratic equations play a crucial role.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is :

where ‘a’, ‘b’, and ‘c’ are coefficients and ‘a’ does not equal zero. This form is crucial for solving quadratic equations and understanding their properties.

Part 2. Quadratic Formula

The quadratic formula is a fundamental method for solving quadratic equations. It gives a basic approach for finding the roots of any quadratic equation. To obtain the quadratic formula, we can complete the square for the general form of the quadratic equation. This process leads us to the well-known formula:

x = [-b ± √(b((2)) – 4ac)]/2a.

Anyone who wants to solve quadratic equations must know how to apply this formula. It is also critical to understand additional quadratic equation ormulas , such as the sum and product of root formulas or the vertex form.

How to Derivate Quadratic Formula

To derive the quadratic formula, we start with the standard form of a quadratic equation: ax^2 + bx + c = 0. The goal is to solve x by isolating it on one side of the equation. Here’s the step-by-step process:

How to Derivate Quadratic Formula	Steps	Note
Step 1: Subtract c from both sides	ax^2 + bx + c = 0ax^2 + bx = -c	We want to set up the left side for completing the square, so we move the constant term ‘c’ to the right side.
Step 2: Divide both sides by a	ax^2 + bx = -cx^2 + (b/a)x = -c/a	Dividing by ‘a’ simplifies the equation and sets us up for the next step.
Step 3: Complete the square	To complete the square, we take half of the coefficient of x (which is b/a), square it, and add it to both sides.x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2	Adding (b/2a)^2 to both sides allows us to create a perfect square trinomial on the left side.
Step 4: Simplify the right side	x^2 + (b/a)x + (b^2/4a^2) = (-c + b^2/4a)/a	We combine like terms on the right side to simplify the equation.
Step 5: Write the left side as a perfect square	The left side is now a perfect square trinomial, which can be written as (x + b/2a)^2. The right side simplifies to (b^2 – 4ac)/(4a^2).(x + b/2a)^2 = (b^2 – 4ac)/(4a^2)	The left side is now a square of a binomial, which makes it easier to solve for x.
Step 6: Take the square root of both sides	x + b/2a = ±√(b^2 – 4ac)/(2a)	Taking the square root of both sides allows us to solve for x. The ± sign indicates that there are two possible solutions.
Step 7: Solve for x	Finally, we subtract b/2a from both sides to isolate x:x = [-b ± √(b^2 – 4ac)]/(2a)	This is the quadratic formula, which gives us the roots of the quadratic equation.

Example:

Let’s derive the quadratic formula for the equation x^2 + 5x + 6 = 0.

1. x^2 + 5x + 6 = 0
2. x^2 + 5x = -6
3. x^2 + 5x + (5/2)^2 = -6 + (5/2)^2
4. x^2 + 5x + 25/4 = -6 + 25/4
5. (x + 5/2)^2 = 1/4
6. x + 5/2 = ±√(1/4)
7. x = -5/2 ± 1/2

So the solutions are x = -2 and x = -3.

This process demonstrates how the quadratic formula is derived and how it can be applied to find the roots of a quadratic equation.

Formulas Related to Quadratic Equations

1. Start with a quadratic equation in the form of $ax^2 + bx + c = 0$. Ensure that $a$, the coefficient of $x^2$, is 1. If it’s not, divide every term by $a$ to make it 1.

2. Move the constant term to the other side of the equation. Do this by adding or subtracting the constant term from both sides of the equation so that you have only the $x^2$ and $x$ terms on one side and the constant on the other.

3. Take half of the coefficient of $x$ (which is $b$) and square it. The coefficient of $x$ at this point should be $b$ because we’ve ensured $a = 1$. So, you take $\frac{b}{2}$ and square it, giving you $\left(\frac{b}{2}\right)^2$.

4. Add this value to both sides of the equation. This step is crucial because it “completes the square” on the left side of the equation.

5. Rewrite the left side of the equation as a perfect square. You do this by writing it in the form $\left(x + \frac{b}{2}\right)^2$.

6. Solve for $x$. This involves taking the square root of both sides, remembering to consider both the positive and negative roots, and then simplifying to find the values of $x$.

Let’s go through an example:

Solve $x^2 + 6x + 5 = 0$ by completing the square.

1. The equation is already in the correct form with $a = 1$.

2. Move the constant term to the other side: $x^2 + 6x = -5$.

3. Take half of the coefficient of $x$ and square it: $\left(\frac{6}{2}\right)^2 = 3^2 = 9$.

4. Add this value to both sides of the equation: $x^2 + 6x + 9 = -5 + 9$.

5. Rewrite the left side as a perfect square: $\left(x + 3\right)^2 = 4$.

6. Solve for $x$: $x + 3 = \pm2$, so $x = -3 \pm 2$. This gives us two solutions: $x = -1$ and $x = -5$.

How to Use the Formula to Solve Quadratic Equations

1. Write down the quadratic equation in its standard form: $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are coefficients and $a \neq 0$.

2. Identify the coefficients: Determine the values of $a$, $b$, and $c$ from the equation.

3. Use the quadratic formula: Substitute the values of $a$, $b$, and $c$ into the quadratic formula, which is $x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$.

4. Calculate the discriminant: The discriminant, $D$, is given by $D = b^2 – 4ac$. Calculate this value first as it will tell you how many real roots the equation has:

   • If $D > 0$, there are two distinct real roots.
   • If $D = 0$, there is exactly one real root (a repeated root).
   • If $D < 0$, there are no real roots, but two complex roots.

5. Solve for $x$: Calculate the two possible values of $x$ using the quadratic formula. Remember to consider both the positive and negative square root of the discriminant to find both solutions.

6. Simplify the solutions: Simplify the expressions under the square root and fraction to find the final values of $x$.

Part 3. How to Solve Quadratic Equations

There are other techniques to solve quadratic equations, but factoring is one of the more natural approaches. We can convert the quadratic equation into two linear factors by identifying two values whose sum is ‘b’ and whose product is ‘c’. However, this approach is only applicable to rationally rooting equations.

Another effective method for turning the equation into a perfect square trinomial is to complete the square. This approach is especially helpful for comprehending how the quadratic formula is derived.

For those who prefer a more direct approach, using a quadratic equation calculator can provide immediate solutions. These calculators can be found online and are a quick way to solve quadratic equations without manual calculations.

How to Solve Quadratic Equations by Factoring

Solving quadratic equations by factoring involves breaking down the quadratic expression into its linear factors and then using the zero product property to find the solutions. Here’s a step-by-step guide:

Step	Description
Standard Form	Start with the quadratic equation in standard form, \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
Factor the Quadratic	Factor the quadratic: Find two binomials that multiply to give you the original quadratic expression. This step can vary in difficulty depending on the coefficients and whether the quadratic easily factors into integers, requires factoring out a greatest common factor first, or involves more complex factoring like difference of squares or perfect square trinomials.
Set Each Factor Equal to Zero	Set each factor equal to zero: Use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, you set each factor equal to zero to create separate linear equations.
Solve Each Linear Equation	Solve each linear equation: Solve each equation for \(x\) to find the roots of the original quadratic equation.

Example 1:

Solve the equation: 2x^2 – 5x -3 =0

Example 2:

Solve the equation: x^2 + 6x + 9= 0

Remember, not all quadratic equations can be easily factored, and in such cases, other methods like completing the square, using the quadratic formula, or graphing may be necessary to find the solutions.

How to Complete the Square

How to Solve Quadratic Equation by Calculators

To solve a quadratic equation using a calculator, follow these steps:

1. Identify the coefficients: For a quadratic equation in the form ax^2+bx+c=0, determine the values of a, b, and c.
2. Select the appropriate mode: If your calculator has different modes (e.g., degree, radian, complex), ensure it’s set to the correct mode for your equation. For real roots, the degree mode is typically used.
3. Use the quadratic formula function: Many scientific calculators have a dedicated function or app for solving quadratic equations. Look for a button or menu option labeled “QUAD” or similar. Select this option.
4. Enter the coefficients: When prompted, enter the values of a, b, and c. Make sure to enter them correctly to get the right solutions.
5. Calculate the solutions: After entering the coefficients, the calculator will process the quadratic formula to find the roots of the equation. It will display the solutions, which can be one or two real roots, or two complex roots if the discriminant is negative.
6. Record the solutions: Write down the solutions provided by the calculator. If the solutions are complex numbers and your question requires real roots, note that the equation has no real roots.
7. Verify the solutions: To ensure accuracy, you can verify the solutions by plugging them back into the original equation and checking if they satisfy the equation.

Quadratic Equation Calculators

Quadratic Equation Calculators	Features
Symbolab Quadratic Equation Calculator	This calculator provides detailed step-by-step solutions, supports solving equations by factoring, completing the square, using the quadratic formula, and even graphing the equations. It’s very user-friendly and offers additional resources for learning.
Mathway Quadratic Equation Calculator	Mathway’s calculator is known for its simplicity and accuracy. It offers solutions using the quadratic formula and also provides a graph of the equation. The interface is straightforward, making it easy for users to input their equations and understand the solutions.
Wolfram|Alpha Quadratic Equation Calculato	Wolfram|Alpha’s calculator is powerful and offers in-depth analysis of the quadratic equations. It provides solutions, explanations of each step, and graphs. The calculator can also handle more complex mathematical problems beyond quadratic equations.

Part 4. Roots of a Quadratic Equation

The roots of a quadratic equation can be either real or imaginary, depending on the discriminant, which is the value b^2 – 4ac. A positive discriminant indicates two distinct real roots, a zero discriminant means one real root, and a negative discriminant results in two complex roots.

The sum and product of the roots of a quadratic equation can be calculated using the coefficients ‘a’, ‘b’, and ‘c’. These relationships are not only useful for solving equations but also for understanding the behavior of quadratic functions.

Nature of Roots of the Quadratic Equation

Here’s the information about the nature of roots of a quadratic equation, keeping the language consistent with the user’s input:

Nature of Roots of the Quadratic Equation

The roots of a quadratic equation can be either real or imaginary, depending on the discriminant, which is the value b^2−44ac. To determine the nature of the roots, you calculate this discriminant based on the coefficients of the quadratic equation ax2+bx+c=0. Here’s how the discriminant affects the roots:

• If the discriminant is positive (b2−4ac>0), the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.
• If the discriminant is zero (b2−4ac=0), the quadratic equation has exactly one real root, which is a repeated root. This corresponds to the parabola just touching the x-axis at a single point.
• If the discriminant is negative (b2−4ac<0), the quadratic equation has two complex roots. These roots are complex numbers, and the parabola does not intersect the x-axis.

Understanding the nature of the roots is crucial for solving quadratic equations and for interpreting the graph of a quadratic function.

• The discriminant value, b2 – 4ac, is used to determine the nature of the roots.

Sum and Product of Roots of Quadratic Equation

The sum and product of the roots of a quadratic equation can be calculated directly from the coefficients of the equation, which are typically represented by ‘a’, ‘b’, and ‘c’. For a quadratic equation in the form ax2+bx+c=0, the sum and product of the roots are determined as follows:

• If p and q are the roots of the quadratic equation, then the sum of the roots is given by p+q=−ab. This relationship shows that the sum of the roots is equal to the negation of the coefficient ‘b’ divided by the coefficient ‘a’.
• The product of the roots, p⋅q, is given by pq=ac. This indicates that the product of the roots is equal to the coefficient ‘c’ divided by the coefficient ‘a’.

These relationships are not only useful for solving quadratic equations but also for understanding the behavior of quadratic functions. By knowing the sum and product of the roots, one can gain insights into the nature of the roots without having to find the roots explicitly. This is particularly useful in algebra and calculus for analyzing the properties of functions and their graphs.

How to Write Quadratic Equations Using Roots

To write a quadratic equation using the roots, you can use the fact that the roots of a quadratic equation in the form ax2+bx+c=0 are given by the quadratic formula x=2a−b±b2−4ac. If the roots are p and q, then the quadratic equation can be written in terms of its roots as follows:

• The sum of the roots is p+q=−ab.
• The product of the roots is p⋅q=ac.

Using these relationships, you can construct a quadratic equation given its roots. For example, if you have the roots p=2 and q=3, you can find the coefficients ‘a’, ‘b’, and ‘c’ as follows:

• The sum of the roots is p+q=2+3=5, so −b/a=5.
• The product of the roots is p⋅q=2⋅3=6, so a/c=6.

Assuming a=1, you would then have b=−5 and c=6. Thus, the quadratic equation with roots p=2 and q=3 would be x^2−5x+6=0.

Remember, the choice of a=1 is arbitrary. You can choose any non-zero value for a, and then determine b and c accordingly. The key is to ensure that the relationships between the coefficients and the roots are maintained as per the properties of quadratic equations.

Quadratic Equations Having Common Roots

When discussing quadratic equations having common roots, it’s important to understand the conditions under which this occurs. For two quadratic equations to have common roots, their coefficients must satisfy specific relationships.

Let’s consider two quadratic equations:

1. a1x^2 + b1x + c1 = 0

2. a2x^2 + b2x + c2 = 0

For these equations to have common roots, the following conditions must be met:

• The ratios of their coefficients must be equal, except for a potential common multiplier. This means that a1/a2=b1/b2=c1/c2, with the possibility of a common factor among them.
• If the equations have exactly the same roots, then one equation can be obtained from the other by multiplying or dividing the coefficients by a constant factor.

It’s also worth noting that if the discriminant of both equations is zero (b^2 – 4ac = 0), and the ratios of their coefficients are equal, then they will have exactly one common real root, which is a repeated root.

In summary, for two quadratic equations to have common roots, their coefficients must be in a specific proportion to each other, allowing for a potential common factor. This relationship ensures that the equations share at least one root, or in the case of identical ratios and a zero discriminant, they share both roots.

Part 5. Graphical Solution of Quadratic Equations

Graphing quadratic equations reveals the parabolic nature of these functions. The shape of the parabola and its orientation (upward or downward) are determined by the coefficient ‘a’. The solutions to the quadratic equation are the points where the parabola intersects the x-axis, if any.

The graphical representation also allows for a geometric interpretation of the maximum or minimum value of the quadratic function, which is located at the vertex of the parabola.

Graphing Quadratic Equations

Graphing quadratic equations involves plotting the points that satisfy the equation y=ax^2+bx+c on a coordinate plane. Here’s how you can graph a quadratic equation:

1. Identify the Vertex: The vertex of a parabola is its highest or lowest point, depending on whether it opens upwards or downwards. The vertex can be found using the formula x=−b/2a for the x-coordinate, and substituting this value back into the equation to find the y-coordinate.
2. Determine the Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. Its equation is x=−b/2a.
3. Find the y-intercept: The y-intercept is the point where the parabola crosses the y-axis. To find it, substitute x=0 into the equation and solve for y.
4. Calculate the Discriminant: The discriminant, D=b^2−4ac, helps determine the nature of the roots, which in turn affects the graph. If D>0, the parabola intersects the x-axis at two points; if D=0, it touches the x-axis at one point; if D<0, it does not intersect the x-axis.
5. Plot Additional Points: Besides the vertex, y-intercept, and x-intercepts (if any), calculate and plot additional points by substituting different x-values into the equation to get corresponding y-values. This helps in sketching the curve more accurately.
6. Sketch the Parabola: Connect the plotted points with a smooth curve to form a parabola. The parabola opens upwards if a>0 and downwards if a<0.

Part 6. Quadratic Equations Worksheet

To reinforce the understanding of quadratic equations, working through a series of problems is invaluable. A quadratic equations worksheet offers a variety of problems that cover different methods of solving and interpreting quadratic equations.

Quadratic Equations Worksheet – Download for Free

Part 7. Quadratic Equations Examples

Problem 1

Problem	Solve the quadratic equation x^2−5x+6=0.
Answer	The roots are x=2 and x=3.
Solution	We can solve this equation by factoring. The equation can be factored as (x−2)(x−3)=0. Setting each factor equal to zero gives us x−2=0 and x−3=0, which yield the solutions x=2 and x=3.

Problem 2

Problem	Find the roots of the equation x^2−3x−10=0.
Answer	The roots are x=−2 and x=5.
Solution	This equation can also be solved by factoring. It factors as (x−5)(x+2)=0. Setting each factor equal to zero gives us x−5=0 and x+2=0, which yield the solutions x=5 and x=−2.

Problem 3

Problem	Solve for x in the equation 2x^2+7x+3=0.
Answer	The roots are x=−1/2 and x=−3.

Solution

Problem 4

Problem	Determine the roots of the equation x^2−6x+9=0.
Answer	The root is x=3 (a repeated root).
Solution	This equation can be solved by factoring. It factors as (x−3)(x−3)=0. Setting each factor equal to zero gives us x−3=0, which yields the repeated solution x=3.

Problem 5

Problem	Solve the quadratic equation 3x^2−10x+3=0.
Answer	The root is x=1/3 and x=3.

Solution

Part 8. Practice Questions on Quadratic Equation

Part 9. Applications and Examples of Quadratic Equations

The quadratic formula is not just a theoretical concept; it has practical applications in various fields. In physics and engineering, for example, quadratic equations are used to model the trajectory of projectiles, the behavior of springs, and other dynamic systems.

In real-world problems, such as calculating the time it takes for an object to hit the ground after being dropped, or determining the dimensions of a rectangular area for maximum efficiency, quadratic equations provide the necessary tools for finding solutions.

Quadratic Formula in Real-World Problems

• Quadratic equations are used in physics, engineering, economics, and computer science to model real-world problems.

Quadratic Equations in Physics and Engineering

• Quadratic equations are used to describe the motion of objects, including the trajectory of projectiles and the vibration of springs.

FAQs about Quadratic Equations

What is discriminant in a quadratic equation?

The discriminant, denoted as D or Δ, is a part of the quadratic formula used to determine the nature of the roots of a quadratic equation. It is calculated as D=b^2−4ac. If D>0, the equation has two distinct real roots; if D=0, it has one real root (a repeated root); and if D<0, it has two complex roots.

How do you graph a quadratic equation?

Graphing a quadratic equation involves plotting the points that satisfy the equation y=ax^2+bx+c on a coordinate plane. The graph of a quadratic equation is a parabola. To graph it, you can find the vertex, determine the axis of symmetry, locate the y-intercept, and calculate additional points to plot. The shape and direction of the parabola depend on the coefficient if a>0, the parabola opens upwards; if a<0, it opens downwards.

Can a quadratic equation have no real roots?

Yes, a quadratic equation can have no real roots. This occurs when the discriminant D=b^2−4ac is negative. In such cases, the equation has two complex roots, which are conjugates of each other.

How do you find the vertex of a parabola?

The vertex of a parabola given by y=ax^2+bx+c can be found using the formula x=−b/2a for the x-coordinate of the vertex. To find the y-coordinate, substitute this value of x back into the equation and solve for y. The vertex represents the maximum or minimum point of the parabola, depending on whether it opens downwards or upwards, respectively.

These FAQs cover fundamental aspects of quadratic equations, from solving and graphing to understanding the nature of their roots.

Summary

The quadratic equation is a pivotal concept in algebra, with far-reaching implications in various fields. This guide has explored the definition, formulas, graphical representations, solutions, and applications of quadratic equations. By understanding the nature of roots, the methods of solving, and the real-world uses of quadratic equations, readers can gain a comprehensive grasp of this essential mathematical tool. Whether you’re solving for ‘x’ or graphing a parabola, the principles outlined here will serve as a solid foundation for all your quadratic equation needs.

If you have any questions about Quadratic Equations, you are welcome to make a free reservation for WuKong Math trial class at any time to get an online live trial class. New users can also receive free online math learning resources.