Write the following equations in factored form. Remember to common factor first
(if possible) 𝑦 = 2𝑥2 + 3𝑥 − 2
First, we can try to common factor the expression inside the brackets:
𝑦 = 2𝑥2 + 3𝑥 − 2
𝑦 = (2𝑥 − 1)(𝑥 + 2)
Therefore, the factored form of the equation is:
𝑦 = (2𝑥 − 1)(𝑥 + 2)
To factor the quadratic equation 𝑦 = 2𝑥^2 + 3𝑥 - 2, we need to look for common factors first.
Step 1: Check for common factors
There are no common factors among the coefficients (2, 3, -2) and the constant term (-2). Therefore, we proceed to the next step.
Step 2: Decompose the middle term
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). In this case, the product is -4 and the sum is 3.
The numbers that satisfy these conditions are 4 and -1. So, we rewrite the middle term 3𝑥 as 4𝑥 - 𝑥:
𝑦 = 2𝑥^2 + 4𝑥 - 𝑥 - 2
Step 3: Group the terms
Group the terms in pairs:
(2𝑥^2 + 4𝑥) + (-𝑥 - 2)
Step 4: Factor by grouping
Factor out the common factors from each pair:
2𝑥(𝑥 + 2) - 1(𝑥 + 2)
Step 5: Combine the common factors
Combine the common factors (𝑥 + 2):
(2𝑥 - 1)(𝑥 + 2)
The factored form of the equation 𝑦 = 2𝑥^2 + 3𝑥 - 2 is (2𝑥 - 1)(𝑥 + 2).