Questions: Factor. 9 x^2+22 x-15 9 x^2+22 x-15 = (Factor completely.)

Transcript text: Factor. \[ 9 x^{2}+22 x-15 \] $9 x^{2}+22 x-15=$ $\square$ (Factor completely.)

Solution

Solution Steps

To factor the quadratic expression $9x^2 + 22x - 15$, we need to find two numbers that multiply to the product of the coefficient of $x^2$ term (9) and the constant term (-15), which is -135, and add up to the coefficient of the $x$ term (22). Once these numbers are found, we can use them to split the middle term and factor by grouping.

Step 1: Identify the Quadratic Expression

We are given the quadratic expression $9x^2 + 22x - 15$.

Step 2: Determine the Product and Sum

To factor the expression, we need two numbers that multiply to the product of the coefficient of $x^2$ (which is 9) and the constant term (which is -15). This product is $9 \times -15 = -135$. We also need these two numbers to add up to the coefficient of $x$, which is 22.

Step 3: Find the Numbers

The numbers that satisfy these conditions are 27 and -5, since $27 \times -5 = -135$ and $27 + (-5) = 22$.

Step 4: Split the Middle Term

Using these numbers, we can split the middle term of the quadratic expression: \[ 9x^2 + 27x - 5x - 15 \]

Step 5: Factor by Grouping

Group the terms to factor by grouping: \[ (9x^2 + 27x) + (-5x - 15) \]

Factor out the greatest common factor from each group: \[ 9x(x + 3) - 5(x + 3) \]

Step 6: Factor Out the Common Binomial

Now, factor out the common binomial $(x + 3)$: \[ (x + 3)(9x - 5) \]