Question
Factor `8x^{2}+16x-10`completely.
Answers
GPT-4o mini
To factor the expression \( 8x^{2} + 16x - 10 \) completely, we can start by looking for a common factor among the terms.
1. **Identify the greatest common factor (GCF)**:
The GCF of the coefficients \( 8, 16, \) and \( -10 \) is \( 2 \). We can factor out \( 2 \):
\[
8x^{2} + 16x - 10 = 2(4x^{2} + 8x - 5)
\]
2. **Factor the quadratic expression**:
Now, we focus on factoring the quadratic \( 4x^{2} + 8x - 5 \). We can do this by looking for two numbers that multiply to \( ac \) (where \( a \) is the coefficient of \( x^{2} \) and \( c \) is the constant term) and add to \( b \) (the coefficient of \( x \)).
Here, \( a = 4 \), \( b = 8 \), and \( c = -5 \).
Thus, \( ac = 4 \times -5 = -20 \) and \( b = 8 \).
We need two numbers that multiply to \( -20 \) and add to \( 8 \). The numbers \( 10 \) and \( -2 \) fit this requirement:
- \( 10 \times (-2) = -20 \)
- \( 10 + (-2) = 8 \)
3. **Rewrite the quadratic**:
We can rewrite the middle term \( 8x \) using these two numbers:
\[
4x^{2} + 10x - 2x - 5
\]
4. **Group the terms**:
Now, we group the terms:
\[
(4x^{2} + 10x) + (-2x - 5)
\]
5. **Factor by grouping**:
Factor out the common factors in each group:
\[
2x(2x + 5) - 1(2x + 5)
\]
This gives us:
\[
(2x - 1)(2x + 5)
\]
6. **Combine with the GCF**:
Now, we combine this back with the GCF we factored out initially:
\[
2(2x - 1)(2x + 5)
\]
Thus, the completely factored form of the expression \( 8x^{2} + 16x - 10 \) is:
\[
\boxed{2(2x - 1)(2x + 5)}
\]
1. **Identify the greatest common factor (GCF)**:
The GCF of the coefficients \( 8, 16, \) and \( -10 \) is \( 2 \). We can factor out \( 2 \):
\[
8x^{2} + 16x - 10 = 2(4x^{2} + 8x - 5)
\]
2. **Factor the quadratic expression**:
Now, we focus on factoring the quadratic \( 4x^{2} + 8x - 5 \). We can do this by looking for two numbers that multiply to \( ac \) (where \( a \) is the coefficient of \( x^{2} \) and \( c \) is the constant term) and add to \( b \) (the coefficient of \( x \)).
Here, \( a = 4 \), \( b = 8 \), and \( c = -5 \).
Thus, \( ac = 4 \times -5 = -20 \) and \( b = 8 \).
We need two numbers that multiply to \( -20 \) and add to \( 8 \). The numbers \( 10 \) and \( -2 \) fit this requirement:
- \( 10 \times (-2) = -20 \)
- \( 10 + (-2) = 8 \)
3. **Rewrite the quadratic**:
We can rewrite the middle term \( 8x \) using these two numbers:
\[
4x^{2} + 10x - 2x - 5
\]
4. **Group the terms**:
Now, we group the terms:
\[
(4x^{2} + 10x) + (-2x - 5)
\]
5. **Factor by grouping**:
Factor out the common factors in each group:
\[
2x(2x + 5) - 1(2x + 5)
\]
This gives us:
\[
(2x - 1)(2x + 5)
\]
6. **Combine with the GCF**:
Now, we combine this back with the GCF we factored out initially:
\[
2(2x - 1)(2x + 5)
\]
Thus, the completely factored form of the expression \( 8x^{2} + 16x - 10 \) is:
\[
\boxed{2(2x - 1)(2x + 5)}
\]
shorten the answer
GPT-4o mini
The completely factored form of \( 8x^{2} + 16x - 10 \) is:
\[
\boxed{2(2x - 1)(2x + 5)}
\]
\[
\boxed{2(2x - 1)(2x + 5)}
\]
put the steps but keep them short
GPT-4o mini
Sure! Here are the condensed steps for factoring \( 8x^{2} + 16x - 10 \):
1. **Factor out GCF**:
\[
8x^{2} + 16x - 10 = 2(4x^{2} + 8x - 5)
\]
2. **Factor the quadratic** \( 4x^{2} + 8x - 5 \):
- Find numbers that multiply to \( -20 \) (product of \( 4 \times -5\)) and add to \( 8 \): these are \( 10 \) and \( -2 \).
3. **Rewrite and group**:
\[
4x^{2} + 10x - 2x - 5 = (4x^{2} + 10x) + (-2x - 5)
\]
4. **Factor by grouping**:
\[
2x(2x + 5) - 1(2x + 5) = (2x - 1)(2x + 5)
\]
5. **Combine with GCF**:
\[
2(2x - 1)(2x + 5)
\]
Final factored form:
\[
\boxed{2(2x - 1)(2x + 5)}
\]
1. **Factor out GCF**:
\[
8x^{2} + 16x - 10 = 2(4x^{2} + 8x - 5)
\]
2. **Factor the quadratic** \( 4x^{2} + 8x - 5 \):
- Find numbers that multiply to \( -20 \) (product of \( 4 \times -5\)) and add to \( 8 \): these are \( 10 \) and \( -2 \).
3. **Rewrite and group**:
\[
4x^{2} + 10x - 2x - 5 = (4x^{2} + 10x) + (-2x - 5)
\]
4. **Factor by grouping**:
\[
2x(2x + 5) - 1(2x + 5) = (2x - 1)(2x + 5)
\]
5. **Combine with GCF**:
\[
2(2x - 1)(2x + 5)
\]
Final factored form:
\[
\boxed{2(2x - 1)(2x + 5)}
\]