(a-3)(a+7)=-9
a(a+7)-3(a+7)=-9
a^2+7a-3a-21+9=0
a^2+4a-12=0
a^2+6a-2a-12=0
a(a+6)-2(a+6)=0
a+6=0
a=-6
OR
a-2=0
a=2
To solve the equation (a-3)(a+7) = -9, we can follow these steps:
Step 1: Expand the equation.
(a-3)(a+7) = -9
(a^2 + 7a - 3a - 21) = -9
Step 2: Combine like terms.
a^2 + 4a - 21 = -9
Step 3: Move all the terms to one side of the equation to create a quadratic equation in standard form.
a^2 + 4a - 21 + 9 = 0
a^2 + 4a - 12 = 0
Step 4: Factor the quadratic equation if possible.
To factor the quadratic equation, we need to find two numbers whose product is -12 and sum is 4.
Let's try different pairs of factors of -12 until we find the correct pair:
1 * -12 = -12
2 * -6 = -12
3 * -4 = -12
The pair that adds up to 4 is 6 and -2:
6 * -2 = -12
6 + (-2) = 4
So, we can rewrite the equation as:
(a + 6)(a - 2) = 0
Step 5: Set each factor equal to zero and solve for a.
a + 6 = 0 --> a = -6
a - 2 = 0 --> a = 2
Thus, the solutions to the equation (a-3)(a+7) = -9 are a = -6 and a = 2.