What Are The Solutions To $2(x-7)^2=32$?A. $x=7 \pm \sqrt{32}$B. $x= \pm \sqrt{65}$C. $x=3$ And $x=11$D. $x=-1$ And $x=15$

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the different methods and techniques used to find the solutions. In this article, we will focus on solving the quadratic equation 2(x−7)2=322(x-7)^2=32 and explore the different solutions that can be obtained.

Understanding the Equation

The given equation is 2(x−7)2=322(x-7)^2=32. To solve this equation, we need to isolate the variable xx. The first step is to expand the squared term using the formula (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2. Applying this formula to the given equation, we get:

2(x2−14x+49)=322(x^2 - 14x + 49) = 32

Expanding and Simplifying the Equation

Now, let's expand and simplify the equation by distributing the coefficient 2 to the terms inside the parentheses:

2x2−28x+98=322x^2 - 28x + 98 = 32

Next, we can subtract 32 from both sides of the equation to get:

2x2−28x+66=02x^2 - 28x + 66 = 0

Solving the Quadratic Equation

To solve the quadratic equation 2x2−28x+66=02x^2 - 28x + 66 = 0, we can use the quadratic formula, which is given by:

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

In this case, a=2a = 2, b=−28b = -28, and c=66c = 66. Plugging these values into the quadratic formula, we get:

x=−(−28)±(−28)2−4(2)(66)2(2)x = \frac{-(-28) \pm \sqrt{(-28)^2 - 4(2)(66)}}{2(2)}

Simplifying the Quadratic Formula

Now, let's simplify the quadratic formula by evaluating the expressions inside the square root:

x=28±784−5284x = \frac{28 \pm \sqrt{784 - 528}}{4}

x=28±2564x = \frac{28 \pm \sqrt{256}}{4}

Evaluating the Square Root

The square root of 256 is 16, so we can simplify the equation further:

x=28±164x = \frac{28 \pm 16}{4}

Finding the Solutions

Now, let's find the two possible solutions by evaluating the expressions inside the square root:

x=28+164=444=11x = \frac{28 + 16}{4} = \frac{44}{4} = 11

x=28−164=124=3x = \frac{28 - 16}{4} = \frac{12}{4} = 3

Conclusion

In conclusion, the solutions to the quadratic equation 2(x−7)2=322(x-7)^2=32 are x=3x=3 and x=11x=11. These solutions can be obtained by using the quadratic formula and simplifying the resulting expressions.

Final Answer

The final answer is: C\boxed{C}

Introduction

Solving quadratic equations can be a challenging task, especially for those who are new to mathematics. In this article, we will address some of the most frequently asked questions about solving quadratic equations, including the equation 2(x−7)2=322(x-7)^2=32 that we solved earlier.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable is two. It is typically written in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

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

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

A: To use the quadratic formula, you need to identify the values of aa, bb, and cc in the quadratic equation. Then, you can plug these values into the quadratic formula and simplify the resulting expression to find the solutions.

Q: What is the difference between the two solutions obtained from the quadratic formula?

A: The two solutions obtained from the quadratic formula are called the distinct roots of the quadratic equation. They are obtained by using the plus and minus signs in the quadratic formula.

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 distinct roots.

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

A: To determine the number of solutions of a quadratic equation, you can use the discriminant, which is the expression inside the square root in the quadratic formula. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is the expression inside the square root in the quadratic formula, which is given by:

b2−4acb^2 - 4ac

Q: How do I simplify the quadratic formula?

A: To simplify the quadratic formula, you can start by evaluating the expressions inside the square root. Then, you can simplify the resulting expression by combining like terms.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with complex coefficients. However, you need to be careful when simplifying the resulting expression, as it may involve complex numbers.

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 identifying the values of aa, bb, and cc correctly
  • Not simplifying the resulting expression correctly
  • Not checking the discriminant to determine the number of solutions
  • Not using the correct signs in the quadratic formula

Conclusion

In conclusion, solving quadratic equations can be a challenging task, but with the right techniques and formulas, you can find the solutions with ease. By understanding the quadratic formula and the discriminant, you can determine the number of solutions of a quadratic equation and simplify the resulting expression to find the solutions.

Final Answer

The final answer is: C\boxed{C}