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?
Answers
GPT-4o mini
To factor the quadratic equation \( 2x^{2} + 7x - 15 = 0 \), we can use the method of finding two numbers that multiply to the product of the coefficient of \( x^{2} \) (which is 2) and the constant term (-15), and also add up to the coefficient of \( x \) (which is 7).
1. Calculate the product:
\[
2 \times -15 = -30
\]
2. We need to find two numbers that multiply to -30 and add to 7. The numbers that satisfy these conditions are 10 and -3.
3. Rewrite the middle term (7x) using these numbers:
\[
2x^{2} + 10x - 3x - 15 = 0
\]
4. Now group the terms:
\[
(2x^{2} + 10x) + (-3x - 15) = 0
\]
5. Factor out the common factors from each group:
\[
2x(x + 5) - 3(x + 5) = 0
\]
6. Now factor out the common binomial \((x + 5)\):
\[
(2x - 3)(x + 5) = 0
\]
Thus, the correctly factored form of the equation \( 2x^{2} + 7x - 15 = 0 \) is:
\[
(2x - 3)(x + 5) = 0
\]
1. Calculate the product:
\[
2 \times -15 = -30
\]
2. We need to find two numbers that multiply to -30 and add to 7. The numbers that satisfy these conditions are 10 and -3.
3. Rewrite the middle term (7x) using these numbers:
\[
2x^{2} + 10x - 3x - 15 = 0
\]
4. Now group the terms:
\[
(2x^{2} + 10x) + (-3x - 15) = 0
\]
5. Factor out the common factors from each group:
\[
2x(x + 5) - 3(x + 5) = 0
\]
6. Now factor out the common binomial \((x + 5)\):
\[
(2x - 3)(x + 5) = 0
\]
Thus, the correctly factored form of the equation \( 2x^{2} + 7x - 15 = 0 \) is:
\[
(2x - 3)(x + 5) = 0
\]