Math

QuestionFactor and solve the equation 6a221a12=06 a^{2}-21 a-12=0. List your answers separated by a comma.

Studdy Solution

STEP 1

Assumptions1. The given equation is 6a21a12=06 a^{}-21 a-12=0 . We are asked to solve this equation by factoring

STEP 2

First, we need to factor the quadratic equation. A quadratic equation is in the form ax2+bx+c=0ax^{2} + bx + c=0. The equation can be factored into two binomial expressions if there exist two numbers m and n such that1. m + n = b (the coefficient of a)
2. m * n = c (the constant term)

STEP 3

In this case, the equation is 6a221a12=06 a^{2}-21 a-12=0. We need to find two numbers that add up to -21 and multiply to -72 (since6*-12 = -72).

STEP 4

The two numbers that satisfy these conditions are -24 and3, because -24 +3 = -21 and -24 *3 = -72.

STEP 5

We rewrite the middle term of the equation as the sum of -24a and3a.
a221a12=0 a^{2}-21 a-12=0becomesa224a+3a12=0 a^{2}-24a+3a-12=0

STEP 6

Now, we factor by grouping. This involves factoring out the greatest common factor (GCF) from each of the two groups.
6a224a+3a12=06 a^{2}-24a+3a-12=0becomes6a(a4)+3(a4)=06a(a-4)+3(a-4)=0

STEP 7

Notice that (a-4) is a common factor. We can factor it out.
6a(a4)+3(a4)=06a(a-4)+3(a-4)=0becomes(a4)(6a+3)=0(a-4)(6a+3)=0

STEP 8

Now, we set each factor equal to zero and solve for a.
(a4)(6a+3)=0(a-4)(6a+3)=0becomesa4=0or6a+3=0a-4=0 \quad or \quad6a+3=0

STEP 9

olving the first equation a4=a-4= for a, we get a=4a=4.

STEP 10

olving the second equation 6a+3=06a+3=0 for a, we get a=2a=-\frac{}{2}.
Therefore, the solutions to the equation 6a221a12=06 a^{2}-21 a-12=0 are a=4a=4 and a=2a=-\frac{}{2}.

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