Question

write the following equations in factored form. Remember to common factor first

(if possible).
𝑦 = 2𝑥power2 + 3𝑥 − 2

Answer 1

[AI-generated answer — Bot]

We first look for common factors, but there are none. Therefore, we proceed to factor the quadratic equation using the quadratic formula or by factoring by grouping:

𝑦 = 2𝑥² + 3𝑥 − 2

Using the quadratic formula:

𝑥 = [−(3) ± √(3² − 4(2)(−2))]/(2(2))
𝑥 = [−3 ± √(25)]/4
𝑥 = [−3 ± 5]/4

Thus, the solutions are:

𝑥 = −1/2 or 𝑥 = 2

Therefore, the factored form of the equation is:

𝑦 = 2(𝑥 + 1/2)(𝑥 − 2)

Answer 2

[AI-generated answer — Step-by-Step Bot]

To factor the equation 𝑦 = 2𝑥^2 + 3𝑥 - 2, we will look for common factors first, and then proceed with factoring if possible.

First, let's check if there is a common factor among all the terms. In this case, there isn't one.

Now, we will attempt to factor the quadratic expression. We need to find two numbers that multiply to give -4 (the coefficient of 𝑥^2 multiplied by the constant term -2) and add up to the coefficient of 𝑥, which is 3.

To find these numbers, let's list all the possible factor pairs of -4:
-4 and 1
-2 and 2

Among these pairs, we can see that -2 and 2 add up to 0, not 3. Thus, we will try the other pair, -4 and 1.

We can rewrite the middle term, 3𝑥, as -4𝑥 + 7𝑥 (since -4𝑥 + 7𝑥 = 3𝑥):
𝑦 = 2𝑥^2 - 4𝑥 + 7𝑥 - 2

Next, we will factor by grouping. We group the first two terms together and the last two terms together:
𝑦 = (2𝑥^2 - 4𝑥) + (7𝑥 - 2)

Now, let's factor out the greatest common factor (GCF) from each group:
𝑦 = 2𝑥(𝑥 - 2) + 1(𝑥 - 2)

Notice that we have a common binomial factor, (𝑥 - 2), in both groups. We can factor it out, and we are left with:
𝑦 = (2𝑥 + 1)(𝑥 - 2)

Therefore, the factored form of the equation 𝑦 = 2𝑥^2 + 3𝑥 - 2 is (2𝑥 + 1)(𝑥 - 2).