Write the following equations in factored form. Remember to common factor first
(if possible).
π¦ = 2π₯power 2 + 3π₯ β 2
First, let's see if we can common factor anything. The coefficients of the quadratic term and the linear term are both even, so we can factor out a 2:
π¦ = 2(π₯power 2 + (3/2)π₯ β 1)
Now we need to factor the quadratic term inside the parentheses. We can use either factoring by grouping or the quadratic formula. For this example, we'll use factoring by grouping:
π¦ = 2(π₯ β 1/2)(π₯ + 2)
So the factored form of the equation π¦ = 2π₯power 2 + 3π₯ β 2 is π¦ = 2(π₯ β 1/2)(π₯ + 2).
To factorize the quadratic equation π¦ = 2π₯Β² + 3π₯ - 2, we'll need to find two binomials that multiply to give us the original equation.
1. Start by looking for a common factor among the terms:
In this case, there is no common factor among the terms.
2. Next, we need to find two numbers that multiply to give π Γ π (in this case, 2 Γ -2) and add up to give π (in this case, 3).
The numbers that satisfy these conditions are 4 and -1.
3. Rewrite the equation using these two numbers as coefficients for the middle term:
π¦ = 2π₯Β² + 4π₯ - π₯ - 2
4. Group the terms:
π¦ = (2π₯Β² + 4π₯) - (π₯ + 2)
5. Factor out the common factors from each group:
π¦ = 2π₯(π₯ + 2) - 1(π₯ + 2)
6. Notice that there is now a common binomial factor of (π₯ + 2). We can factor that out:
π¦ = (π₯ + 2)(2π₯ - 1)
Therefore, the factored form of the equation π¦ = 2π₯Β² + 3π₯ - 2 is (π₯ + 2)(2π₯ - 1).