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

We'll start with factoring:

1. Identify $a$, $b$, and $c$ in the trinomial $a{x}^2+bx+c$:

   • $a=9$
   • $b=-9$
   • $c=-4$

2. Calculate the discriminant $D=b^2-4ac$: $$D=(-9{)}^2-4(9)(-4)=81+144=225$$ Since the discriminant is a perfect square, we can factor it.

3. Find the roots using the quadratic formula: $$x=\frac{-b\pm \sqrt{D}}{2a}=\frac{9\pm 15}{18}$$

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

4. Write the factored form using the roots: The factors corresponding to the roots are $x-\frac{4}{3}$ and $x+\frac{1}{3}$.

5. Adjusting the factors to eliminate the fractions: Multiply each factor by 3: $$3(x-\frac{4}{3})=3x-4,3(x+\frac{1}{3})=3x+1$$

Thus, the factored form becomes: $$(3x-4)(3x+1)$$

Putting this in the requested format, the answer is: $(3x-4)(3x+1)$