b How do you solve a quadratic equation by factoring?
C
c How do you solve a quadratic equation by factoring?
A
aExample Solution: x^2-8x+15=0
B
bExample Solution: x^2+x-6=0
C
cExample Solution: x^2 +7x +6=0
Practice makes perfect
a Recall than we can solve a quadratic equation by factoring and using the Zero Product Property. We can do the inverse process to find a quadratic equation with the solutions that we want. We start with the product of two binomials of the form (x-c)(x-d)=0, where c and d are real numbers.
(x-c)(x-d)=0
Zero Product Property
a* b = 0 ⇔ a =0 or b=0
x-c =0 1.5cm x-d=0
or
x=c 2.05cm x=d
We can see that the product (x-c)(x-d) will be zero when x=c or x=d. Since we want the solution of our equations to be 3 and 5, we can use these values for c and d and then expand the product of the binomial to obtain a quadratic expression.
Therefore, the equation x^2-8x+15 = 0 has the solutions 3 and 5, as required. However, it is important to state that this is not the only solution satisfying this problem's requirements, nor the only way to construct them. There are infinitely many equations satisfying this problem, and this is only an example solution.
b We can follow the same process we used in Part A. This time we want our solutions to be -3 and 2. This time, we will expand the product (x-(-3))(x-2).
Therefore, the equation x^2+x-6=0 has the solutions -3 and 2, as required. Again, notice that this is is not the only possible solution, and it is just an example solution.
c Once more we can follow the method described in Part A, now using the values -1 and -6, and expanding the product on the left-hand side of the equation (x-(-1))(x-(-6))=0.
