Question
7.
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.
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.
Answers
GPT-4o mini
To factor the quadratic expression \( 8x^2 + 16x - 10 \) completely, we can follow these steps:
### Step 1: Factor out the greatest common factor (GCF)
First, we need to identify the GCF of the coefficients. The coefficients are 8, 16, and -10, and the GCF is 2.
Factoring out 2 from the expression, we get:
\[
2(4x^2 + 8x - 5)
\]
### Step 2: Factor the quadratic expression \( 4x^2 + 8x - 5 \)
Now we need to factor the quadratic \( 4x^2 + 8x - 5 \). We can use the quadratic formula to find its roots, or we can try to factor it directly.
The quadratic is in the form \( ax^2 + bx + c \), where:
- \( a = 4 \)
- \( b = 8 \)
- \( c = -5 \)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
### Step 3: Calculate the discriminant
Now, we need to find the discriminant:
\[
b^2 - 4ac = 8^2 - 4(4)(-5) = 64 + 80 = 144
\]
### Step 4: Apply the quadratic formula
Now we can find the roots:
\[
x = \frac{-8 \pm \sqrt{144}}{2(4)} = \frac{-8 \pm 12}{8}
\]
This gives us two roots:
1. \( x = \frac{4}{8} = \frac{1}{2} \)
2. \( x = \frac{-20}{8} = -\frac{5}{2} \)
### Step 5: Write the factored form
Now we can express \( 4x^2 + 8x - 5 \) using its roots:
\[
4x^2 + 8x - 5 = 4(x - \frac{1}{2})(x + \frac{5}{2})
\]
### Step 6: Simplify the factored form
To express the quadratic factor in a more standard form, we can write:
\[
4(x - \frac{1}{2})(x + \frac{5}{2}) = (4x - 2)(x + \frac{5}{2})
\]
### Step 7: Combine everything
Now we combine everything:
\[
8x^2 + 16x - 10 = 2(4x - 2)(x + \frac{5}{2})
\]
So, \( 8x^2 + 16x - 10 = 2(4x - 2)(x + \frac{5}{2}) \).
### Final Factored Expression
Thus, the final factored form of the original expression is:
\[
2(4x - 2)(x + \frac{5}{2})
\]
### Identifying a Factor
Therefore, one of the factors of \( 8x^2 + 16x - 10 \) is:
\[
4x - 2
\]
You can also present the other factor and the GCF, but the question specifically asks for one of the factors.
### Step 1: Factor out the greatest common factor (GCF)
First, we need to identify the GCF of the coefficients. The coefficients are 8, 16, and -10, and the GCF is 2.
Factoring out 2 from the expression, we get:
\[
2(4x^2 + 8x - 5)
\]
### Step 2: Factor the quadratic expression \( 4x^2 + 8x - 5 \)
Now we need to factor the quadratic \( 4x^2 + 8x - 5 \). We can use the quadratic formula to find its roots, or we can try to factor it directly.
The quadratic is in the form \( ax^2 + bx + c \), where:
- \( a = 4 \)
- \( b = 8 \)
- \( c = -5 \)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
### Step 3: Calculate the discriminant
Now, we need to find the discriminant:
\[
b^2 - 4ac = 8^2 - 4(4)(-5) = 64 + 80 = 144
\]
### Step 4: Apply the quadratic formula
Now we can find the roots:
\[
x = \frac{-8 \pm \sqrt{144}}{2(4)} = \frac{-8 \pm 12}{8}
\]
This gives us two roots:
1. \( x = \frac{4}{8} = \frac{1}{2} \)
2. \( x = \frac{-20}{8} = -\frac{5}{2} \)
### Step 5: Write the factored form
Now we can express \( 4x^2 + 8x - 5 \) using its roots:
\[
4x^2 + 8x - 5 = 4(x - \frac{1}{2})(x + \frac{5}{2})
\]
### Step 6: Simplify the factored form
To express the quadratic factor in a more standard form, we can write:
\[
4(x - \frac{1}{2})(x + \frac{5}{2}) = (4x - 2)(x + \frac{5}{2})
\]
### Step 7: Combine everything
Now we combine everything:
\[
8x^2 + 16x - 10 = 2(4x - 2)(x + \frac{5}{2})
\]
So, \( 8x^2 + 16x - 10 = 2(4x - 2)(x + \frac{5}{2}) \).
### Final Factored Expression
Thus, the final factored form of the original expression is:
\[
2(4x - 2)(x + \frac{5}{2})
\]
### Identifying a Factor
Therefore, one of the factors of \( 8x^2 + 16x - 10 \) is:
\[
4x - 2
\]
You can also present the other factor and the GCF, but the question specifically asks for one of the factors.