rite the following equations in factored form. Remember to common factor first
(if possible).
π¦ = 2π₯power2 + 3π₯ β 2
To factor this quadratic equation, we first look for a common factor of the quadratic and linear terms. The greatest common factor here is 1, so we move on to finding two numbers whose product is -4 and whose sum is 3. These numbers are 4 and -1. We can now rewrite the quadratic in factored form:
π¦ = (2π₯ - 1)(π₯ + 2)
To factor the equation π¦ = 2π₯^2 + 3π₯ - 2, we need to find common factors, if possible.
First, let's look for any common factors among the coefficients of the terms. The coefficients 2, 3, and -2 do not share any common factors other than 1. So, we can't factor out any common factors from the coefficients.
Next, we will try to factor the quadratic expression. To do that, we need to find two numbers whose product is equal to the product of the coefficient of the squared term (2) and the constant term (-2), which is -4, and whose sum is equal to the coefficient of the linear term (3).
After trying different combinations, we find that the numbers 4 and -1 satisfy these conditions. So, we can express the middle term (3π₯) as the sum of 4π₯ and -π₯.
Now, we can rewrite the equation factored form as follows:
π¦ = 2π₯^2 + 4π₯ - π₯ - 2
Next, we group the terms:
π¦ = (2π₯^2 + 4π₯) - (π₯ + 2)
Now, we factor out the greatest common factor from each group:
π¦ = 2π₯(π₯ + 2) - 1(π₯ + 2)
Finally, we can factor out the binomial π₯ + 2:
π¦ = (2π₯ - 1)(π₯ + 2)
Therefore, the factored form of the equation π¦ = 2π₯^2 + 3π₯ - 2 is (2π₯ - 1)(π₯ + 2).