Question
Factor `8x^{2}+16x-10`completely. Which of the following expressions is a factor?
Answers
GPT-4o mini
To factor the quadratic expression \(8x^2 + 16x - 10\) completely, we'll start by looking for common factors and then apply the factoring process.
1. First, identify the greatest common factor (GCF) of the coefficients in the expression. The GCF of \(8\), \(16\), and \(-10\) is \(2\).
2. Factor out \(2\) from the expression:
\[
8x^2 + 16x - 10 = 2(4x^2 + 8x - 5)
\]
3. Now, we need to factor the quadratic \(4x^2 + 8x - 5\). We can use the quadratic formula or look for factors that could work. We will check if it can be factored.
To do that, we can use the method of trial and error for factoring \(4x^2 + 8x - 5\). We look for two numbers that multiply to \(4 \times -5 = -20\) and add to \(8\). The numbers \(10\) and \(-2\) satisfy this condition because:
\[
10 \cdot (-2) = -20 \quad \text{and} \quad 10 + (-2) = 8
\]
4. Rewrite the middle term using these numbers:
\[
4x^2 + 10x - 2x - 5
\]
5. Now, group the terms:
\[
(4x^2 + 10x) + (-2x - 5)
\]
6. Factor each group:
\[
2x(2x + 5) - 1(2x + 5)
\]
7. Factor out the common binomial factor:
\[
(2x + 5)(2x - 1)
\]
8. Now, we can include the \(2\) we factored out earlier:
\[
2(2x + 5)(2x - 1)
\]
Thus, the completely factored form of the original expression \(8x^2 + 16x - 10\) is:
\[
2(2x + 5)(2x - 1)
\]
From this, we can see that possible factors include \(2\), \(2x + 5\), and \(2x - 1\). The question asks for one of the factors; therefore, one of the factors is \(2x + 5\) or \(2x - 1\).
To conclude, the expression \(2x + 5\) or \(2x - 1\) (the individual factors) can be listed as an answer depending on your needs.
1. First, identify the greatest common factor (GCF) of the coefficients in the expression. The GCF of \(8\), \(16\), and \(-10\) is \(2\).
2. Factor out \(2\) from the expression:
\[
8x^2 + 16x - 10 = 2(4x^2 + 8x - 5)
\]
3. Now, we need to factor the quadratic \(4x^2 + 8x - 5\). We can use the quadratic formula or look for factors that could work. We will check if it can be factored.
To do that, we can use the method of trial and error for factoring \(4x^2 + 8x - 5\). We look for two numbers that multiply to \(4 \times -5 = -20\) and add to \(8\). The numbers \(10\) and \(-2\) satisfy this condition because:
\[
10 \cdot (-2) = -20 \quad \text{and} \quad 10 + (-2) = 8
\]
4. Rewrite the middle term using these numbers:
\[
4x^2 + 10x - 2x - 5
\]
5. Now, group the terms:
\[
(4x^2 + 10x) + (-2x - 5)
\]
6. Factor each group:
\[
2x(2x + 5) - 1(2x + 5)
\]
7. Factor out the common binomial factor:
\[
(2x + 5)(2x - 1)
\]
8. Now, we can include the \(2\) we factored out earlier:
\[
2(2x + 5)(2x - 1)
\]
Thus, the completely factored form of the original expression \(8x^2 + 16x - 10\) is:
\[
2(2x + 5)(2x - 1)
\]
From this, we can see that possible factors include \(2\), \(2x + 5\), and \(2x - 1\). The question asks for one of the factors; therefore, one of the factors is \(2x + 5\) or \(2x - 1\).
To conclude, the expression \(2x + 5\) or \(2x - 1\) (the individual factors) can be listed as an answer depending on your needs.