Let's solve each problem step by step.
---
Problem 1: Which of the following is equivalent to \(\sqrt{112}\)?
We need to simplify \(\sqrt{112}\).
1. Factorize 112 into its prime factors:
\[
112 = 2^4 \times 7
\]
2. Rewrite the square root using the factorization:
\[
\sqrt{112} = \sqrt{2^4 \times 7} = \sqrt{(2^2)^2 \times 7} = 2^2 \sqrt{7} = 4\sqrt{7}
\]
Thus, the correct answer is:
\[
\boxed{D}
\]
---
Problem 2: What are the values of \(a\), \(b\), and \(c\) in the quadratic equation \(3x^2 - 10 = -4x\)?
First, rewrite the equation in standard quadratic form \(ax^2 + bx + c = 0\):
1. Start with the given equation:
\[
3x^2 - 10 = -4x
\]
2. Add \(4x\) to both sides to move all terms to one side:
\[
3x^2 + 4x - 10 = 0
\]
Now, identify the coefficients:
- \(a = 3\)
- \(b = 4\)
- \(c = -10\)
Thus, the correct answer is:
\[
\boxed{A}
\]
---
Problem 3: What is the positive solution to the equation \(0 = \left(\frac{1}{5}\right)x^2 - 5\)?
Solve the equation step by step:
1. Start with the given equation:
\[
0 = \left(\frac{1}{5}\right)x^2 - 5
\]
2. Add 5 to both sides:
\[
5 = \left(\frac{1}{5}\right)x^2
\]
3. Multiply both sides by 5 to isolate \(x^2\):
\[
25 = x^2
\]
4. Take the square root of both sides:
\[
x = \pm 5
\]
Since we are asked for the positive solution:
\[
x = 5
\]
Thus, the correct answer is:
\[
\boxed{C}
\]
---
Problem 4: What are the solutions to \(5(x - 3)^2 = 45\)?
Solve the equation step by step:
1. Start with the given equation:
\[
5(x - 3)^2 = 45
\]
2. Divide both sides by 5:
\[
(x - 3)^2 = 9
\]
3. Take the square root of both sides:
\[
x - 3 = \pm 3
\]
4. Solve for \(x\) in both cases:
- If \(x - 3 = 3\):
\[
x = 6
\]
- If \(x - 3 = -3\):
\[
x = 0
\]
Thus, the solutions are \(x = 6\) and \(x = 0\).
The correct answer is:
\[
\boxed{C}
\]
---
Problem 5: What are the solutions for the equation \(2x^2 + 16x = 18\)?
Solve the equation step by step:
1. Start with the given equation:
\[
2x^2 + 16x = 18
\]
2. Move all terms to one side to set the equation to zero:
\[
2x^2 + 16x - 18 = 0
\]
3. Simplify by dividing the entire equation by 2:
\[
x^2 + 8x - 9 = 0
\]
4. Factor the quadratic equation:
\[
x^2 + 8x - 9 = (x + 9)(x - 1) = 0
\]
5. Set each factor to zero and solve for \(x\):
- If \(x + 9 = 0\):
\[
x = -9
\]
- If \(x - 1 = 0\):
\[
x = 1
\]
Thus, the solutions are \(x = -9\) and \(x = 1\).
The correct answer is:
\[
\boxed{B}
\]
---
Problem 6: The area of a certain rhombus is given by the equation \(L^2 - 5L = 6\), where \(L\) represents the length of the rhombus in centimeters. What is the length of the rhombus?
Solve the equation step by step:
1. Start with the given equation:
\[
L^2 - 5L = 6
\]
2. Move all terms to one side to set the equation to zero:
\[
L^2 - 5L - 6 = 0
\]
3. Factor the quadratic equation:
\[
L^2 - 5L - 6 = (L - 6)(L + 1) = 0
\]
4. Set each factor to zero and solve for \(L\):
- If \(L - 6 = 0\):
\[
L = 6
\]
- If \(L + 1 = 0\):
\[
L = -1
\]
Since the length \(L\) cannot be negative, we discard \(L = -1\).
Thus, the length of the rhombus is:
\[
L = 6
\]
The correct answer is:
\[
\boxed{B}
\]
---
Final Answers:
1. \(\boxed{D}\)
2. \(\boxed{A}\)
3. \(\boxed{C}\)
4. \(\boxed{C}\)
5. \(\boxed{B}\)
6. \(\boxed{B}\)
Parent Tip: Review the logic above to help your child master the concept of quadratic formula worksheet algebra 2.
