Question
Dorothy was solving the following quadratic equation by factoring: 2 "x" squared plus 7 "x" minus 15 equals 02x2+7x−15=0. Which choice shows the correctly factored form of this equation?
Answers
GPT-4o mini
To factor the quadratic equation \(2x^2 + 7x - 15 = 0\), we need to find two numbers that multiply to \(2 \times -15 = -30\) and add up to \(7\).
The numbers that satisfy this are \(10\) and \(-3\).
Now, we can rewrite the middle term (\(7x\)) using these two numbers:
\[
2x^2 + 10x - 3x - 15 = 0
\]
Next, we can factor by grouping:
Group the first two terms and the last two terms:
\[
(2x^2 + 10x) + (-3x - 15) = 0
\]
Factor out the common factors in each group:
\[
2x(x + 5) - 3(x + 5) = 0
\]
Now, we can factor out the common binomial factor \((x + 5)\):
\[
(2x - 3)(x + 5) = 0
\]
Thus, the correctly factored form of the equation \(2x^2 + 7x - 15 = 0\) is:
\[
(2x - 3)(x + 5) = 0
\]
The numbers that satisfy this are \(10\) and \(-3\).
Now, we can rewrite the middle term (\(7x\)) using these two numbers:
\[
2x^2 + 10x - 3x - 15 = 0
\]
Next, we can factor by grouping:
Group the first two terms and the last two terms:
\[
(2x^2 + 10x) + (-3x - 15) = 0
\]
Factor out the common factors in each group:
\[
2x(x + 5) - 3(x + 5) = 0
\]
Now, we can factor out the common binomial factor \((x + 5)\):
\[
(2x - 3)(x + 5) = 0
\]
Thus, the correctly factored form of the equation \(2x^2 + 7x - 15 = 0\) is:
\[
(2x - 3)(x + 5) = 0
\]