Solve 9r^2-36r-45

$$9 { r }^{ 2 } -36r-45$$

Factor

$$9\left(r-5\right)\left(r+1\right)$$

Solution Steps

Factor out 9.

$$9\left(r^{2}-4r-5\right)$$

Consider r^{2}-4r-5. Factor the expression by grouping. First, the expression needs to be rewritten as r^{2}+ar+br-5. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-5\right)=-5$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-5 b=1$$

Rewrite r^{2}-4r-5 as \left(r^{2}-5r\right)+\left(r-5\right).

$$\left(r^{2}-5r\right)+\left(r-5\right)$$

Factor out r in r^{2}-5r.

$$r\left(r-5\right)+r-5$$

Factor out common term r-5 by using distributive property.

$$\left(r-5\right)\left(r+1\right)$$

Rewrite the complete factored expression.

$$9\left(r-5\right)\left(r+1\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$9r^{2}-36r-45=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$r=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 9\left(-45\right)}}{2\times 9}$$

Square -36.

$$r=\frac{-\left(-36\right)±\sqrt{1296-4\times 9\left(-45\right)}}{2\times 9}$$

Multiply -4 times 9.

$$r=\frac{-\left(-36\right)±\sqrt{1296-36\left(-45\right)}}{2\times 9}$$

Multiply -36 times -45.

$$r=\frac{-\left(-36\right)±\sqrt{1296+1620}}{2\times 9}$$

Add 1296 to 1620.

$$r=\frac{-\left(-36\right)±\sqrt{2916}}{2\times 9}$$

Take the square root of 2916.

$$r=\frac{-\left(-36\right)±54}{2\times 9}$$

The opposite of -36 is 36.

$$r=\frac{36±54}{2\times 9}$$

Multiply 2 times 9.

$$r=\frac{36±54}{18}$$

Now solve the equation r=\frac{36±54}{18} when ± is plus. Add 36 to 54.

$$r=\frac{90}{18}$$

Divide 90 by 18.

$$r=5$$

Now solve the equation r=\frac{36±54}{18} when ± is minus. Subtract 54 from 36.

$$r=-\frac{18}{18}$$

Divide -18 by 18.

$$r=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and -1 for x_{2}.

$$9r^{2}-36r-45=9\left(r-5\right)\left(r-\left(-1\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$9r^{2}-36r-45=9\left(r-5\right)\left(r+1\right)$$

Evaluate

$$9\left(r-5\right)\left(r+1\right)$$