write the following equations in factored form. Remember to common factor first
(if possible).
𝑦 = 2𝑥2 + 3𝑥 − 2
First, we look for a common factor. In this case, we can factor out 2:
𝑦 = 2𝑥2 + 3𝑥 − 2
𝑦 = 2(𝑥2 + (3/2)𝑥 − 1)
Next, we need to factor the quadratic expression inside the parentheses. We can use the quadratic formula or factoring by grouping:
𝑥2 + (3/2)𝑥 − 1 = (𝑥 + 1)(𝑥 − 1/2)
Substituting this back into the original equation, we get:
𝑦 = 2(𝑥 + 1)(𝑥 − 1/2)
To factor the equation 𝑦 = 2𝑥^2 + 3𝑥 − 2, we need to look for common factors first.
The coefficients of the terms 2𝑥^2, 3𝑥, and -2 are all divisible by 1, so there is no common factor besides 1. Therefore, we can proceed directly to factoring the quadratic trinomial.
To factor the quadratic trinomial, we need to find two numbers that multiply to give the product of the leading coefficient (2) and the constant term (-2) and add up to the coefficient of the middle term (3).
The product of 2 and -2 is -4, and the two numbers that multiply to give -4 and add up to 3 are 4 and -1. So, we can rewrite the equation as follows:
𝑦 = (𝑥 + 4)(𝑥 − 1)
Therefore, the factored form of the equation 𝑦 = 2𝑥^2 + 3𝑥 − 2 is 𝑦 = (𝑥 + 4)(𝑥 − 1).