To factor the quadratic expression \( 8x^2 + 16x - 10 \) completely, we follow these steps:
-
Factor out the greatest common factor (GCF): The coefficients of the terms are \( 8 \), \( 16 \), and \( -10 \). The GCF of these numbers is \( 2 \). So we can start by factoring out \( 2 \):
\[ 8x^2 + 16x - 10 = 2(4x^2 + 8x - 5) \]
-
Factor the quadratic: Now we need to factor the quadratic expression \( 4x^2 + 8x - 5 \). We will look for two numbers that multiply to \( ac \) (where \( a = 4 \) and \( c = -5 \)), which is \( 4 \times -5 = -20 \), and add up to \( b = 8 \).
The pairs of factors of \( -20 \) that add up to \( 8 \) are:
- \( 10 \) and \( -2 \)
So we can rewrite \( 4x^2 + 8x - 5 \) using these numbers:
\[ 4x^2 + 10x - 2x - 5 \]
Now we can group the terms:
\[ (4x^2 + 10x) + (-2x - 5) \]
Factor out the common factors in each group:
\[ 2x(2x + 5) - 1(2x + 5) \]
Now we can factor out \( (2x + 5) \):
\[ (2x + 5)(2x - 1) \]
-
Combine everything: We can now combine this back with the factor of \( 2 \) we factored out earlier:
\[ 8x^2 + 16x - 10 = 2(2x + 5)(2x - 1) \]
The completely factored form of \( 8x^2 + 16x - 10 \) is:
\[ 2(2x + 5)(2x - 1) \]
- Check the options: Now, we need to see which of the given expressions is a factor of the quadratic expression:
- a: \( x - 10 \) (not a factor)
- b: \( 2x + 3 \) (not a factor)
- c: \( 4x - 1 \) (not in the factorization)
- d: \( 2x + 5 \) (is a factor)
The factor from the original expression is \((2x + 5)\), so the correct answer is option d: \( 2x + 5 \).
3 answers