(3) Which Value Of $x$ Is A Solution To The Equation $4x^2 + 26x + 12 = 0$?A. -0.5 B. -6 C. -2.5 D. -4.5

Feb 28, 2025 by ADMIN 112 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $4x^2 + 26x + 12 = 0$ , and determine which value of $x$ is a solution to the equation.

Understanding Quadratic Equations

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

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

This formula can be used to find the solutions to any quadratic equation.

Applying the Quadratic Formula

Now, let's apply the quadratic formula to the given equation, $4x^2 + 26x + 12 = 0$ . We have:

$a = 4$ , $b = 26$ , and $c = 12$

Substituting these values into the quadratic formula, we get:

x = \frac{-26 \pm \sqrt{26^2 - 4(4)(12)}}{2(4)}

Simplifying the Expression

Simplifying the expression under the square root, we get:

x = \frac{-26 \pm \sqrt{676 - 192}}{8}

x = \frac{-26 \pm \sqrt{484}}{8}

x = \frac{-26 \pm 22}{8}

Finding the Solutions

Now, we have two possible solutions:

x = \frac{-26 + 22}{8} = \frac{-4}{8} = -0.5

x = \frac{-26 - 22}{8} = \frac{-48}{8} = -6

Conclusion

In this article, we have solved the quadratic equation $4x^2 + 26x + 12 = 0$ using the quadratic formula. We have found two possible solutions, $x = -0.5$ and $x = -6$ . However, we need to determine which of these values is a solution to the equation.

Checking the Solutions

To check the solutions, we can substitute each value back into the original equation and see if it is true.

For $x = -0.5$ , we get:

4(-0.5)^2 + 26(-0.5) + 12 = 1 - 13 + 12 = 0

This is true, so $x = -0.5$ is a solution to the equation.

For $x = -6$ , we get:

4(-6)^2 + 26(-6) + 12 = 144 - 156 + 12 = 0

This is also true, so $x = -6$ is a solution to the equation.

Final Answer

Therefore, the value of $x$ that is a solution to the equation $4x^2 + 26x + 12 = 0$ is both $x = -0.5$ and $x = -6$ . However, since the question only asks for one value, we can choose either one as the final answer.

The final answer is: $\boxed{-0.5}$

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Introduction

In our previous article, we solved the quadratic equation $4x^2 + 26x + 12 = 0$ using the quadratic formula. In this article, we will answer some 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 is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The quadratic formula is a powerful tool for solving quadratic equations and is given by:

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

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to find the solutions to any quadratic equation. It is given by:

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

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to substitute the values of $a$ , $b$ , and $c$ into the formula. For example, if you have the quadratic equation $x^2 + 5x + 6 = 0$ , you can substitute $a = 1$ , $b = 5$ , and $c = 6$ into the formula.

Q: What is the difference between the two solutions of a quadratic equation?

A: The two solutions of a quadratic equation are given by the quadratic formula:

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

The two solutions are:

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

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

The two solutions are different because of the plus or minus sign in front of the square root.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can only have two solutions. This is because the quadratic formula always gives two solutions, and there is no way to get more than two solutions from the quadratic formula.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. This happens when the discriminant, $b^2 - 4ac$ , is negative. In this case, the quadratic formula will give complex solutions, which are not real numbers.

Q: How do I check if a value is a solution to a quadratic equation?

A: To check if a value is a solution to a quadratic equation, you can substitute the value into the equation and see if it is true. For example, if you have the quadratic equation $x^2 + 5x + 6 = 0$ and you want to check if $x = 2$ is a solution, you can substitute $x = 2$ into the equation and see if it is true.

Conclusion

In this article, we have answered some frequently asked questions about quadratic equations. We have discussed the quadratic formula, how to apply it, and how to check if a value is a solution to a quadratic equation. We hope that this article has been helpful in answering your questions about quadratic equations.

Final Tips

Always check your work when solving a quadratic equation.
Make sure to substitute the correct values into the quadratic formula.
Use the quadratic formula to find the solutions to a quadratic equation.
Check if a value is a solution to a quadratic equation by substituting it into the equation.

We hope that this article has been helpful in your understanding of quadratic equations. If you have any further questions, please don't hesitate to ask.