Question

Factor `8x^{2}+16x-10`completely. Which of the following expressions is a factor?



You will need to show your work to get full credit on this problem. You can either show it on the sketch pad to the left, or attach your own document on the next screen.

Answer 1

To factor the quadratic expression \(8x^{2}+16x-10\), we can use the following steps:

1. **Identify the coefficients** in our quadratic expression \(ax^2 + bx + c\):
- \(a = 8\)
- \(b = 16\)
- \(c = -10\)

2. **Factor out the greatest common factor (GCF)**:
The GCF of the coefficients \(8, 16,\) and \(-10\) is \(2\). We factor \(2\) out of the entire expression:

\[
8x^2 + 16x - 10 = 2(4x^2 + 8x - 5)
\]

3. **Now, we need to factor the quadratic \(4x^2 + 8x - 5\)**. We will use the method of factoring by grouping or the AC method.

- To apply the AC method, we compute \(A \times C\), where \(A = 4\) and \(C = -5\):
\[
AC = 4 \times (-5) = -20
\]

- We need to find two numbers that multiply to \(-20\) and add up to \(B = 8\). The numbers \(10\) and \(-2\) meet this criterion because:
\[
10 \times (-2) = -20 \quad \text{and} \quad 10 + (-2) = 8
\]

4. **Rewrite the expression** using these two numbers:
\[
4x^2 + 10x - 2x - 5
\]

5. **Group the terms**:
\[
(4x^2 + 10x) + (-2x - 5)
\]

6. **Factor by grouping**:
- From the first group, we can factor out \(2x\):
\[
2x(2x + 5)
\]
- From the second group, we can factor out \(-1\):
\[
-1(2x + 5)
\]

Now we can combine both parts:
\[
2x(2x + 5) - 1(2x + 5)
\]
\[
= (2x - 1)(2x + 5)
\]

7. **Combine everything**: Don't forget the \(2\) we factored out initially:
\[
8x^2 + 16x - 10 = 2(2x - 1)(2x + 5)
\]

8. **Conclusion**: The completely factored form of \(8x^{2}+16x-10\) is:
\[
2(2x - 1)(2x + 5)
\]

One of the factors of the original expression is \(2x - 1\) or \(2x + 5\). So, you can choose either as a valid factor.