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:
- \( x = \frac{4}{8} = \frac{1}{2} \)
- \( x = \frac{-20}{8} = -\frac{5}{2} \)
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.
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