Factor the trinomial completely.

Question

Type your answers to each problem in the table below. The order of the factors doesn't matter, but use parentheses around each one!

9x^2 - 9x - 4

Answer 1

To factor the trinomial $9x^2-9x-4$, we can use the method of factoring by grouping or applying the quadratic formula to find the roots first.

1. Identify $a=9$, $b=-9$, and $c=-4$.

2. Calculate the discriminant: $$D=b^2-4ac=(-9{)}^2-4(9)(-4)=81+144=225$$

3. The square root of the discriminant: $$\sqrt{D}=\sqrt{225}=15$$

4. Now use the quadratic formula to find the roots: $$x=\frac{-b\pm \sqrt{D}}{2a}=\frac{-(-9)\pm 15}{2(9)}=\frac{9\pm 15}{18}$$

   • For $x_1=\frac{24}{18}=\frac{4}{3}$
   • For $x_2=\frac{-6}{18}=-\frac{1}{3}$

5. Now we can write the trinomial in its factored form based on its roots: $$9x^2-9x-4=9(x-\frac{4}{3})(x+\frac{1}{3})$$

6. To express it without fractions, multiply the factors by 3: $$=(3x-4)(3x+1)$$

So, the factored form is: $$(3x-4)(3x+1)$$

Here is how you can summarize it in the table format you mentioned:

| Factor 1 | Factor 2 | |----------|------------| | (3x - 4) | (3x + 1) |