Which Is (are) The Root(s) Of − 12 X 2 = 60 X + 75 -12x^2 = 60x + 75 − 12 X 2 = 60 X + 75 ?A. X = − 2.5 X = -2.5 X = − 2.5 B. X = ± 2.5 X = \pm 2.5 X = ± 2.5 C. X = 0 , 3 X = 0, 3 X = 0 , 3 D. X = 3 , − 2.5 X = 3, -2.5 X = 3 , − 2.5

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore the process of solving quadratic equations, with a focus on finding the roots of the equation $-12x^2 = 60x + 75$. We will examine the different methods for solving quadratic equations, including factoring, the quadratic formula, and graphing. By the end of this article, you will have a clear understanding of how to find the roots of a quadratic equation and be able to apply this knowledge to a variety of problems.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The roots of a quadratic equation are the values of $x$ that satisfy the equation, and they can be found using various methods.

The Quadratic Formula

One of the most common methods for solving quadratic equations is the quadratic formula. The quadratic formula is given by:

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

This formula can be used to find the roots of any quadratic equation, regardless of whether it can be factored or not.

Applying the Quadratic Formula to $-12x^2 = 60x + 75$

To find the roots of the equation $-12x^2 = 60x + 75$, we can start by rewriting the equation in the standard form $ax^2 + bx + c = 0$. We can do this by subtracting $60x$ from both sides of the equation and adding $75$ to both sides:

$-12x^2 - 60x + 75 = 0$

Now that we have the equation in standard form, we can use the quadratic formula to find the roots. Plugging in the values of $a$, $b$, and $c$, we get:

$x = \frac{-(-60) \pm \sqrt{(-60)^2 - 4(-12)(75)}}{2(-12)}$

Simplifying the expression, we get:

$x = \frac{60 \pm \sqrt{3600 + 3600}}{-24}$

$x = \frac{60 \pm \sqrt{7200}}{-24}$

$x = \frac{60 \pm 84.85}{-24}$

Now, we can find the two possible values of $x$ by plugging in the positive and negative values of the square root:

$x = \frac{60 + 84.85}{-24}$

$x = \frac{144.85}{-24}$

$x = -6.04$

$x = \frac{60 - 84.85}{-24}$

$x = \frac{-24.85}{-24}$

$x = 1.03$

Finding the Roots of the Equation

Now that we have found the two possible values of $x$, we can determine the roots of the equation. The roots of the equation are the values of $x$ that satisfy the equation, and they can be found by setting the equation equal to zero and solving for $x$. In this case, we have found two possible values of $x$, which are $-6.04$ and $1.03$.

Conclusion

In this article, we have explored the process of solving quadratic equations, with a focus on finding the roots of the equation $-12x^2 = 60x + 75$. We have examined the different methods for solving quadratic equations, including factoring, the quadratic formula, and graphing. By applying the quadratic formula to the equation, we have found the two possible values of $x$, which are $-6.04$ and $1.03$. These values represent the roots of the equation, and they can be used to solve a variety of problems.

Answer

Based on the calculations above, the correct answer is:

B. $x = \pm 2.5$

However, this is not the exact answer we found. We found that the roots of the equation are $-6.04$ and $1.03$, which are not equal to $\pm 2.5$. Therefore, the correct answer is not among the options provided.

Discussion

The equation $-12x^2 = 60x + 75$ is a quadratic equation, and solving it requires the use of the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the roots of any quadratic equation, regardless of whether it can be factored or not. In this article, we have applied the quadratic formula to the equation and found the two possible values of $x$, which are $-6.04$ and $1.03$. These values represent the roots of the equation, and they can be used to solve a variety of problems.

Final Answer

The final answer is not among the options provided. However, we can use the quadratic formula to find the roots of the equation, and the correct answer is:

x = -6.04, 1.03

This is the exact answer we found by applying the quadratic formula to the equation.

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Introduction

In our previous article, we explored the process of solving quadratic equations, with a focus on finding the roots of the equation $-12x^2 = 60x + 75$. We applied the quadratic formula to the equation and found the two possible values of $x$, which are $-6.04$ and $1.03$. In this article, we will answer some common questions related to quadratic equations and provide additional examples to help you understand the concept better.

Q&A

Q: What is the quadratic formula?

A: 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 roots of any quadratic equation, regardless of whether it can be factored or not.

Q: How do I apply the quadratic formula to a quadratic equation?

A: To apply the quadratic formula to a quadratic equation, you need to identify the values of $a$, $b$, and $c$ in the equation. Then, you can plug these values into the quadratic formula and simplify the expression to find the roots of the equation.

Q: What are the roots of a quadratic equation?

A: The roots of a quadratic equation are the values of $x$ that satisfy the equation. They can be found by setting the equation equal to zero and solving for $x$.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, the quadratic formula can be used to solve any quadratic equation, regardless of whether it can be factored or not.

Q: What if the quadratic formula gives me two complex roots?

A: If the quadratic formula gives you two complex roots, it means that the equation has no real solutions. Complex roots are roots that involve the imaginary unit $i$, which is defined as the square root of $-1$.

Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex coefficients. However, you need to be careful when working with complex numbers, as they can be tricky to handle.

Q: What if I have a quadratic equation that can be factored?

A: If you have a quadratic equation that can be factored, you can use factoring to find the roots of the equation. Factoring involves expressing the quadratic equation as a product of two binomials.

Q: Can I use the quadratic formula to solve quadratic equations with rational coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with rational coefficients. Rational coefficients are coefficients that are rational numbers, such as integers or fractions.

Q: What if I have a quadratic equation that has no real solutions?

A: If you have a quadratic equation that has no real solutions, it means that the equation has complex roots. In this case, you can use the quadratic formula to find the complex roots of the equation.

Additional Examples

Example 1: Solving a Quadratic Equation with Rational Coefficients

Solve the quadratic equation $x^2 + 5x + 6 = 0$ using the quadratic formula.

Solution

To solve the quadratic equation $x^2 + 5x + 6 = 0$, we can use the quadratic formula:

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

In this case, $a = 1$, $b = 5$, and $c = 6$. Plugging these values into the quadratic formula, we get:

$x = \frac{-5 \pm \sqrt{5^2 - 4(1)(6)}}{2(1)}$

$x = \frac{-5 \pm \sqrt{25 - 24}}{2}$

$x = \frac{-5 \pm \sqrt{1}}{2}$

$x = \frac{-5 \pm 1}{2}$

Therefore, the roots of the equation are $x = -3$ and $x = -2$.

Example 2: Solving a Quadratic Equation with Complex Coefficients

Solve the quadratic equation $x^2 + 2ix + 1 = 0$ using the quadratic formula.

Solution

To solve the quadratic equation $x^2 + 2ix + 1 = 0$, we can use the quadratic formula:

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

In this case, $a = 1$, $b = 2i$, and $c = 1$. Plugging these values into the quadratic formula, we get:

$x = \frac{-2i \pm \sqrt{(2i)^2 - 4(1)(1)}}{2(1)}$

$x = \frac{-2i \pm \sqrt{-4 - 4}}{2}$

$x = \frac{-2i \pm \sqrt{-8}}{2}$

$x = \frac{-2i \pm 2i\sqrt{2}}{2}$

$x = \frac{-2i \pm 2i\sqrt{2}}{2}$

Therefore, the roots of the equation are $x = -i + i\sqrt{2}$ and $x = -i - i\sqrt{2}$.

Conclusion

In this article, we have answered some common questions related to quadratic equations and provided additional examples to help you understand the concept better. We have also explored the process of solving quadratic equations using the quadratic formula and factoring. By applying the quadratic formula to a quadratic equation, you can find the roots of the equation, which are the values of $x$ that satisfy the equation.