CPM Homework Help : A2C Problem 9-171

Generally, when you are asked to factor, it is understood that you are only to use integers in your factors. If you are allowed to use irrational or complex number, any quadratic can be factored.

By setting the polynomial equal to zero and solving the quadratic equation, you can work backwards to “force factor” any quadratic. Use the solutions of the corresponding quadratic equation to write each of the following expressions as a product of two linear factors.

$x^{2} − 10$
Hint (a):
Treat this as a difference of squares.
Answer (a):
$(x+\sqrt{10})(x-\sqrt{10})$

$x^{2} − 3x − 7$
Hint (b):
Use the Quadratic Formula, then use those answers as factors.
Answer (b):
$\left(x-\frac{3+\sqrt{37}}{2}\right)\left(x-\frac{3-\sqrt{37}}{2}\right)$

$x^{2} + 4$
Hint (c):
Using complex numbers, this can be treated as a difference of squares.
More Help (c):
$x^{2} + 4 = x^{2} − \left(−4\right)$
Answer (c):
$\left(x + 2i\right)\left(x − 2i\right)$

$x^{2} − 2x + 2$
Hint (d):
See part (b).