Verification using Graphing

Oct 13, 2025 by miodeya.com 15 views

Introduction

The quadratic equation is a fundamental concept in algebra, and it has numerous applications in various fields such as physics, engineering, and economics. In this article, we will focus on solving the quadratic equation $3x^2 + 12x - 135 = 0$ using the Quadratic Formula. We will also verify our result by graphing and using the Intersect Method.

Step 1: Write the Equation in Standard Form

The first step in solving a quadratic equation is to write it in standard form, which is $ax^2 + bx + c = 0$ . In this case, the equation is already in standard form, with $a = 3$ , $b = 12$ , and $c = -135$ .

Step 2: Identify the Coefficients $a$ , $b$ , and $c$

The coefficients $a$ , $b$ , and $c$ are the numbers in front of the $x^2$ , $x$ , and constant terms, respectively. In this case, we have:

$a = 3$
$b = 12$
$c = -135$

Step 3: Plug the Coefficients into the Quadratic Formula

The Quadratic Formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

We will plug the values of $a$ , $b$ , and $c$ into this formula to get:

$x = \frac{-12 \pm \sqrt{12^2 - 4(3)(-135)}}{2(3)}$

Step 4: Simplify the Expression

We will simplify the expression inside the square root:

$12^2 - 4(3)(-135) = 144 + 1620 = 1764$

So, the expression becomes:

$x = \frac{-12 \pm \sqrt{1764}}{6}$

Step 5: Simplify the Square Root

The square root of 1764 is 42, so we can simplify the expression as:

$x = \frac{-12 \pm 42}{6}$

Step 6: Solve for $x$

We will solve for $x$ by considering both the plus and minus cases:

$x = \frac{-12 + 42}{6} = \frac{30}{6} = 5$
$x = \frac{-12 - 42}{6} = \frac{-54}{6} = -9$

Therefore, the solutions to the quadratic equation are $x = 5$ and $x = -9$ .

Verification using Graphing

To verify our result, we can graph the quadratic equation using a graphing calculator or software. We will graph the equation $y = 3x^2 + 12x - 135$ and find the x-intercepts, which are the points where the graph crosses the x-axis.

Verification using the Intersect Method

The Intersect Method involves finding the points of intersection between two lines. In this case, we will find the points of intersection between the graph of the quadratic equation and the x-axis.

Conclusion

In this article, we solved the quadratic equation $3x^2 + 12x - 135 = 0$ using the Quadratic Formula. We verified our result by graphing and using the Intersect Method. The solutions to the quadratic equation are $x = 5$ and $x = -9$ .

Final Answer

The final answer is $\boxed{5, -9}$ .

References

[1] "Quadratic Formula" by Math Open Reference
[2] "Graphing Quadratic Equations" by Math Is Fun
[3] "Intersect Method" by Wolfram MathWorld
Quadratic Equation Q&A ==========================

Frequently Asked Questions about Quadratic Equations

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It is in the form of $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

Factoring: If the quadratic expression can be factored into the product of two binomials, we can set each binomial equal to zero and solve for x.
Quadratic Formula: The Quadratic Formula is a formula that can be used to solve any quadratic equation. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Graphing: We can graph the quadratic equation and find the x-intercepts, which are the points where the graph crosses the x-axis.

Q: What is the Quadratic Formula?

A: The Quadratic Formula is a formula that can be used to solve any quadratic equation. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Q: How do I use the Quadratic Formula?

A: To use the Quadratic Formula, we need to plug in the values of $a$ , $b$ , and $c$ into the formula. We then simplify the expression inside the square root and solve for x.

Q: What is the difference between the plus and minus cases in the Quadratic Formula?

A: The plus and minus cases in the Quadratic Formula represent two possible solutions to the quadratic equation. The plus case gives us one solution, while the minus case gives us another solution.

Q: How do I verify my solution using graphing?

A: To verify your solution using graphing, you can graph the quadratic equation and find the x-intercepts, which are the points where the graph crosses the x-axis. If the x-intercepts match the solutions you found using the Quadratic Formula, then your solution is correct.

Q: How do I verify my solution using the Intersect Method?

A: To verify your solution using the Intersect Method, you can find the points of intersection between the graph of the quadratic equation and the x-axis. If the points of intersection match the solutions you found using the Quadratic Formula, then your solution is correct.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

Not simplifying the expression inside the square root
Not using the correct values of $a$ , $b$ , and $c$
Not considering both the plus and minus cases in the Quadratic Formula
Not verifying the solution using graphing or the Intersect Method

Q: How do I apply quadratic equations in real-life situations?

A: Quadratic equations have numerous applications in real-life situations, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.

Conclusion

In this article, we answered some frequently asked questions about quadratic equations. We covered topics such as solving quadratic equations, the Quadratic Formula, graphing, and the Intersect Method. We also discussed some common mistakes to avoid and how to apply quadratic equations in real-life situations.