Find The Solution(s) To $x^2 - 16x + 64 = 0$.A. $x = 8$ And $x = -8$ B. $x = -2$ And $x = 32$ C. $x = 8$ Only D. $x = 4$ And $x = 16$

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, $x^2 - 16x + 64 = 0$, and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means 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, and $a$ cannot be zero.

The Quadratic Equation $x^2 - 16x + 64 = 0$

The given quadratic equation is:

$$x^2 - 16x + 64 = 0$$

To solve this equation, we can use various methods, including factoring, completing the square, and the quadratic formula.

Factoring

One way to solve the quadratic equation is by factoring. We can rewrite the equation as:

$$(x - 8)^2 = 0$$

This can be factored as:

$$(x - 8)(x - 8) = 0$$

Using the zero-product property, we can set each factor equal to zero and solve for $x$:

$$x - 8 = 0 \Rightarrow x = 8$$

Since the two factors are the same, we only have one solution, $x = 8$.

Completing the Square

Another method to solve the quadratic equation is by completing the square. We can rewrite the equation as:

$$x^2 - 16x + 64 = (x - 8)^2$$

Expanding the right-hand side, we get:

$$x^2 - 16x + 64 = x^2 - 16x + 64$$

This equation is true for all values of $x$, so we can conclude that the solutions are all real numbers.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

In our case, $a = 1$, $b = -16$, and $c = 64$. Plugging these values into the formula, we get:

$$x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(64)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{16 \pm \sqrt{256 - 256}}{2}$$

This simplifies to:

$$x = \frac{16 \pm \sqrt{0}}{2}$$

Since the square root of zero is zero, we have:

$$x = \frac{16}{2}$$

This simplifies to:

$$x = 8$$

Conclusion

In this article, we have solved the quadratic equation $x^2 - 16x + 64 = 0$ using various methods, including factoring, completing the square, and the quadratic formula. We have found that the solutions are $x = 8$.

Answer

The correct answer is:

• A. $x = 8$ and $x = -8$

This is because the equation can be factored as $(x - 8)^2 = 0$, which has two solutions, $x = 8$ and $x = -8$.

Final Thoughts

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Introduction

In our previous article, we solved the quadratic equation $x^2 - 16x + 64 = 0$ using various methods, including factoring, completing the square, and the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations and provide additional insights and examples.

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 (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, and $a$ cannot be zero.

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 factor equal to zero and solve for $x$.
• Completing the square: We can rewrite the quadratic expression in the form $(x - h)^2 + k$, where $h$ and $k$ are constants.
• The quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

Q: What is the difference between a quadratic equation and a linear equation?

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (in this case, $x$) is one. The general form of a linear equation is:

$$ax + b = 0$$

where $a$ and $b$ are constants, and $a$ cannot be zero.

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

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula always produces two solutions, and factoring and completing the square also produce two solutions.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. This occurs when the discriminant ($b^2 - 4ac$) is negative.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, we can use the discriminant ($b^2 - 4ac$). If the discriminant is:

• Positive: The equation has two distinct solutions.
• Zero: The equation has one repeated solution.
• Negative: The equation has no real solutions.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. This occurs when the discriminant ($b^2 - 4ac$) is negative.

Q: How do I find the complex solutions to a quadratic equation?

A: To find the complex solutions to a quadratic equation, we can use the quadratic formula:

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

If the discriminant is negative, we can rewrite the expression under the square root as:

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

where $i$ is the imaginary unit.

Conclusion

In this article, we have answered some frequently asked questions about quadratic equations and provided additional insights and examples. We have discussed the different methods for solving quadratic equations, including factoring, completing the square, and the quadratic formula. We have also explored the properties of quadratic equations, including the number of solutions and the existence of complex solutions.