Solve For $c$.$(c+9)^2=64$A. $ C = 17 , C = 1 C=17, C=1 C = 17 , C = 1 [/tex] B. $c=-\sqrt{73}, C=\sqrt{73}$ C. $c=-17, C=-1$ D. $c=-\sqrt{55}, C=\sqrt{55}$

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 focus on solving a specific type of quadratic equation, namely the equation $(c+9)^2=64$. We will break down the solution step by step, and provide a clear explanation of each step.

Understanding the Equation

The given equation is $(c+9)^2=64$. To solve for $c$, we need to isolate $c$ on one side of the equation. The first step is to expand the left-hand side of the equation using the formula $(a+b)^2=a^2+2ab+b^2$.

Expanding the Equation

Using the formula, we can expand the left-hand side of the equation as follows:

$$(c+9)^2=c^2+2(c)(9)+9^2$$

Simplifying the equation, we get:

$$c^2+18c+81=64$$

Simplifying the Equation

Now, we can simplify the equation by subtracting 64 from both sides:

$$c^2+18c+17=0$$

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

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

In this case, $a=1$, $b=18$, and $c=17$. Plugging these values into the formula, we get:

$$c=\frac{-18\pm\sqrt{18^2-4(1)(17)}}{2(1)}$$

Simplifying the equation, we get:

$$c=\frac{-18\pm\sqrt{324-68}}{2}$$

$$c=\frac{-18\pm\sqrt{256}}{2}$$

$$c=\frac{-18\pm16}{2}$$

Finding the Solutions

Now, we can find the two possible solutions for $c$:

$$c_1=\frac{-18+16}{2}=-1$$

$$c_2=\frac{-18-16}{2}=-17$$

Conclusion

In this article, we solved the quadratic equation $(c+9)^2=64$ step by step. We expanded the left-hand side of the equation, simplified it, and then used the quadratic formula to find the two possible solutions for $c$. The solutions are $c=-17$ and $c=-1$.

Answer

The correct answer is:

• C. $c=-17, c=-1$

Discussion

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Introduction

In our previous article, we solved the quadratic equation $(c+9)^2=64$ step by step. We expanded the left-hand side of the equation, simplified it, and then used the quadratic formula to find the two possible solutions for $c$. In this article, we will answer some frequently asked questions about quadratic equations and solving for $c$.

Q: What is a quadratic equation?

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

Q: How do I solve a quadratic equation?

There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The method you choose will depend on the specific equation and the values of the coefficients.

Q: What is the quadratic formula?

The quadratic formula is a formula that can be used to solve quadratic equations of the form $ax^2+bx+c=0$. It is given by:

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

Q: How do I use the quadratic formula?

To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the equation. Then, you plug these values into the formula and simplify. The formula will give you two possible solutions for $x$.

Q: What are the two possible solutions for $c$?

In the equation $(c+9)^2=64$, we found two possible solutions for $c$: $c=-17$ and $c=-1$. These solutions are obtained by using the quadratic formula and simplifying the equation.

Q: How do I know which solution is correct?

To determine which solution is correct, you need to plug the solutions back into the original equation and check if they satisfy the equation. If they do, then they are the correct solutions.

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

Yes, the quadratic formula can be used to solve any quadratic equation of the form $ax^2+bx+c=0$. However, you need to make sure that the coefficients $a$, $b$, and $c$ are real numbers and that the discriminant $b^2-4ac$ is non-negative.

Q: What is the discriminant?

The discriminant is the expression $b^2-4ac$ that appears in the quadratic formula. It is used to determine the nature of the solutions of the quadratic equation.

Q: How do I determine the nature of the solutions?

The nature of the solutions can be determined by the value of the discriminant. If the discriminant is positive, then the equation has two distinct real solutions. If the discriminant is zero, then the equation has one real solution. If the discriminant is negative, then the equation has no real solutions.

Conclusion

In this article, we answered some frequently asked questions about quadratic equations and solving for $c$. We discussed the quadratic formula, how to use it, and how to determine the nature of the solutions. We also provided examples of how to solve quadratic equations using the quadratic formula.

Answer

The correct answers to the questions are:

• Q: What is a quadratic equation? A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two.
• Q: How do I solve a quadratic equation? There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square.
• Q: What is the quadratic formula? The quadratic formula is a formula that can be used to solve quadratic equations of the form $ax^2+bx+c=0$.
• Q: How do I use the quadratic formula? To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the equation, plug these values into the formula, and simplify.
• Q: What are the two possible solutions for $c$? In the equation $(c+9)^2=64$, we found two possible solutions for $c$: $c=-17$ and $c=-1$.
• Q: How do I know which solution is correct? To determine which solution is correct, you need to plug the solutions back into the original equation and check if they satisfy the equation.
• Q: Can I use the quadratic formula to solve any quadratic equation? Yes, the quadratic formula can be used to solve any quadratic equation of the form $ax^2+bx+c=0$.
• Q: What is the discriminant? The discriminant is the expression $b^2-4ac$ that appears in the quadratic formula.
• Q: How do I determine the nature of the solutions? The nature of the solutions can be determined by the value of the discriminant.