Question
Dorothy was solving the following quadratic equation by factoring:
2x^2 + 7x - 15 = 0. Which choice shows the correctly factored form of this equation?
a (2x - 3)(x + 5) = 0
b (2x - 3)(x - 5) = 0
c (5x - 3) (x + 2) = 0
d (5x - 3)(x - 2) = 0
2x^2 + 7x - 15 = 0. Which choice shows the correctly factored form of this equation?
a (2x - 3)(x + 5) = 0
b (2x - 3)(x - 5) = 0
c (5x - 3) (x + 2) = 0
d (5x - 3)(x - 2) = 0
Answers
GPT-4o mini
To factor the quadratic equation \(2x^2 + 7x - 15 = 0\), we need to find two numbers that multiply to \(2 \cdot -15 = -30\) and add up to \(7\).
The pairs of factors of \(-30\) that add to \(7\) are \(10\) and \(-3\). Therefore, we can rewrite the middle term \(7x\) as \(10x - 3x\):
\[
2x^2 + 10x - 3x - 15 = 0
\]
Next, we can group the terms:
\[
(2x^2 + 10x) + (-3x - 15) = 0
\]
Factoring out the common factors from each group:
\[
2x(x + 5) - 3(x + 5) = 0
\]
Now we can factor out the common binomial factor \((x + 5)\):
\[
(2x - 3)(x + 5) = 0
\]
The correctly factored form of the equation is:
\[
(2x - 3)(x + 5) = 0
\]
Thus, the correct choice is:
**a (2x - 3)(x + 5) = 0**
The pairs of factors of \(-30\) that add to \(7\) are \(10\) and \(-3\). Therefore, we can rewrite the middle term \(7x\) as \(10x - 3x\):
\[
2x^2 + 10x - 3x - 15 = 0
\]
Next, we can group the terms:
\[
(2x^2 + 10x) + (-3x - 15) = 0
\]
Factoring out the common factors from each group:
\[
2x(x + 5) - 3(x + 5) = 0
\]
Now we can factor out the common binomial factor \((x + 5)\):
\[
(2x - 3)(x + 5) = 0
\]
The correctly factored form of the equation is:
\[
(2x - 3)(x + 5) = 0
\]
Thus, the correct choice is:
**a (2x - 3)(x + 5) = 0**