Solve For $x$ In The Equation $x^2 - 14x + 31 = 63$.A. $ X = − 16 X = -16 X = − 16 [/tex] Or $x = 2$B. $x = -7 \pm 3\sqrt{7}$C. $ X = − 2 X = -2 X = − 2 [/tex] Or $x = 16$D. $x = 7 \pm

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Introduction

In this article, we will delve into solving a quadratic equation of the form $x^2 - 14x + 31 = 63$. The goal is to find the value of $x$ that satisfies the given equation. We will use algebraic techniques to manipulate the equation and isolate the variable $x$.

Step 1: Rearrange the Equation

The first step is to rearrange the equation to set it equal to zero. This is done by subtracting 63 from both sides of the equation.

x2−14x+31=63x^2 - 14x + 31 = 63

x2−14x+31−63=63−63x^2 - 14x + 31 - 63 = 63 - 63

x2−14x−32=0x^2 - 14x - 32 = 0

Step 2: Use the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In our case, $a = 1$, $b = -14$, and $c = -32$. Plugging these values into the formula, we get:

x=−(−14)±(−14)2−4(1)(−32)2(1)x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(-32)}}{2(1)}

x=14±196+1282x = \frac{14 \pm \sqrt{196 + 128}}{2}

x=14±3242x = \frac{14 \pm \sqrt{324}}{2}

x=14±182x = \frac{14 \pm 18}{2}

Step 3: Simplify the Solutions

Now that we have the solutions in the form $x = \frac{14 \pm 18}{2}$, we can simplify them by evaluating the two possible values of $x$.

x=14+182=322=16x = \frac{14 + 18}{2} = \frac{32}{2} = 16

x=14−182=−42=−2x = \frac{14 - 18}{2} = \frac{-4}{2} = -2

Conclusion

In conclusion, the solutions to the equation $x^2 - 14x + 31 = 63$ are $x = 16$ and $x = -2$. These values satisfy the original equation and can be verified by plugging them back into the equation.

Final Answer

The final answer is: C\boxed{C}

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Introduction

In our previous article, we solved the quadratic equation $x^2 - 14x + 31 = 63$ and found the solutions to be $x = 16$ and $x = -2$. In this article, we will address some common questions and concerns that readers may have about solving this equation.

Q: What is the quadratic formula and how is it used?

A: The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The quadratic formula is used by plugging in the values of $a$, $b$, and $c$ into the formula and simplifying the expression.

Q: Why do we need to set the equation equal to zero?

A: Setting the equation equal to zero is a necessary step in solving quadratic equations. By doing so, we can use the quadratic formula to find the solutions to the equation.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic expression as a product of two binomials, while the quadratic formula involves using a formula to find the solutions to the equation.

Q: Can we use the quadratic formula to solve all quadratic equations?

A: Yes, the quadratic formula can be used to solve all quadratic equations of the form $ax^2 + bx + c = 0$. However, it may not always be the most efficient method, especially for equations that can be easily factored.

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 setting the equation equal to zero
  • Not plugging in the correct values of $a$, $b$, and $c$
  • Not simplifying the expression correctly
  • Not checking the solutions to make sure they satisfy the original equation

Q: How do we check the solutions to make sure they satisfy the original equation?

A: To check the solutions, we can plug them back into the original equation and make sure that they satisfy the equation. This can be done by substituting the solutions into the equation and simplifying the expression.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

  • Modeling the trajectory of a projectile
  • Finding the maximum or minimum value of a function
  • Solving problems involving optimization
  • Modeling population growth or decline

Conclusion

In conclusion, solving the equation $x^2 - 14x + 31 = 63$ requires a clear understanding of the quadratic formula and how to use it to find the solutions to the equation. By following the steps outlined in this article, readers should be able to solve similar equations and apply the concepts to real-world problems.

Final Answer

The final answer is: C\boxed{C}