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).
-
Calculate the product: \[ 2 \times -15 = -30 \]
-
We need to find two numbers that multiply to -30 and add to 7. The numbers that satisfy these conditions are 10 and -3.
-
Rewrite the middle term (7x) using these numbers: \[ 2x^{2} + 10x - 3x - 15 = 0 \]
-
Now group the terms: \[ (2x^{2} + 10x) + (-3x - 15) = 0 \]
-
Factor out the common factors from each group: \[ 2x(x + 5) - 3(x + 5) = 0 \]
-
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 answer