Which Shows The Correct Substitution Of The Values { A, B,$}$ And { C$}$ From The Equation { -2 = -x + X^2 - 4$}$ Into The Quadratic Formula?Quadratic Formula: ${x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$A.

Introduction

Quadratic equations are a fundamental concept in mathematics, and the quadratic formula is a powerful tool for solving them. In this article, we will explore the correct substitution of values from a given equation into the quadratic formula. We will examine the equation ${-2 = -x + x^2 - 4}$ and determine which option shows the correct substitution of the values ${a, b,}$ and ${c}$ into the quadratic formula.

The Quadratic Formula

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ${ax^2 + bx + c = 0}$. The formula is given by:

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

Substituting Values into the Quadratic Formula

To solve the equation ${-2 = -x + x^2 - 4}$, we need to substitute the values of ${a, b,}$ and ${c}$ into the quadratic formula. Let's examine the options:

Option A

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

${x = \frac{1 \pm \sqrt{1 + 16}}{2}}$

${x = \frac{1 \pm \sqrt{17}}{2}}$

Option B

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

${x = \frac{1 \pm \sqrt{1 + 16}}{2}}$

${x = \frac{1 \pm \sqrt{17}}{2}}$

Option C

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

${x = \frac{1 \pm \sqrt{1 + 16}}{2}}$

${x = \frac{1 \pm \sqrt{17}}{2}}$

Option D

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

${x = \frac{1 \pm \sqrt{1 + 16}}{2}}$

${x = \frac{1 \pm \sqrt{17}}{2}}$

Which Option is Correct?

After examining the options, we can see that all of them show the correct substitution of the values ${a, b,}$ and ${c}$ into the quadratic formula. However, we need to determine which option is the most accurate.

Step-by-Step Solution

To solve the equation ${-2 = -x + x^2 - 4}$, we need to follow these steps:

1. Rearrange the equation to the standard form ${ax^2 + bx + c = 0}$.
2. Identify the values of ${a, b,}$ and ${c}$.
3. Substitute the values into the quadratic formula.
4. Simplify the expression to find the solutions.

Rearranging the Equation

Let's rearrange the equation to the standard form:

${-2 = -x + x^2 - 4}$

${x^2 - x - 2 = 0}$

Identifying the Values

We can see that the values are:

${a = 1}$ ${b = -1}$ ${c = -2}$

Substituting the Values

Now, we can substitute the values into the quadratic formula:

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

${x = \frac{1 \pm \sqrt{1 + 8}}{2}}$

${x = \frac{1 \pm \sqrt{9}}{2}}$

${x = \frac{1 \pm 3}{2}}$

Simplifying the Expression

We can simplify the expression to find the solutions:

${x = \frac{1 + 3}{2}}$

${x = \frac{4}{2}}$

${x = 2}$

${x = \frac{1 - 3}{2}}$

${x = \frac{-2}{2}}$

${x = -1}$

Conclusion

In conclusion, the correct substitution of the values ${a, b,}$ and ${c}$ from the equation ${-2 = -x + x^2 - 4}$ into the quadratic formula is:

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

${x = \frac{1 \pm \sqrt{1 + 8}}{2}}$

${x = \frac{1 \pm \sqrt{9}}{2}}$

${x = \frac{1 \pm 3}{2}}$

This is the most accurate option, and it shows the correct substitution of the values into the quadratic formula.

Final Answer

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Introduction

The quadratic formula is a powerful tool for solving quadratic equations. However, it can be a bit confusing, especially for those who are new to it. In this article, we will answer some of the most frequently asked questions about the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ${ax^2 + bx + c = 0}$. The formula 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, you need to follow these steps:

1. Rearrange the equation to the standard form ${ax^2 + bx + c = 0}$.
2. Identify the values of ${a, b,}$ and ${c}$.
3. Substitute the values into the quadratic formula.
4. Simplify the expression to find the solutions.

Q: What are the values of a, b, and c?

A: The values of ${a, b,}$ and ${c}$ are the coefficients of the quadratic equation. For example, in the equation ${x^2 - 3x - 4 = 0}$, the values are:

${a = 1}$ ${b = -3}$ ${c = -4}$

Q: How do I simplify the expression?

A: To simplify the expression, you need to follow these steps:

1. Simplify the square root expression.
2. Simplify the fraction.
3. Simplify any other expressions.

Q: What are the solutions to the quadratic equation?

A: The solutions to the quadratic equation are the values of ${x}$ that satisfy the equation. You can find the solutions by simplifying the expression and solving for ${x}$.

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

A: Yes, you can use the quadratic formula to solve any quadratic equation of the form ${ax^2 + bx + c = 0}$. However, you need to make sure that the equation is in the standard form and that the values of ${a, b,}$ and ${c}$ are correct.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

• Not rearranging the equation to the standard form.
• Not identifying the values of ${a, b,}$ and ${c}$.
• Not simplifying the expression correctly.
• Not solving for ${x}$ correctly.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with complex solutions. However, you need to be careful when simplifying the expression and solving for ${x}$.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By following the steps outlined in this article, you can use the quadratic formula to solve any quadratic equation of the form ${ax^2 + bx + c = 0}$. Remember to avoid common mistakes and to be careful when simplifying the expression and solving for ${x}$.

Final Answer

The final answer is: $\boxed{Yes}$