Solve -4r^2=18r+18

$$- 4 r ^ { 2 } = 18 r + 18$$

Solve for r

$$r = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$r=-3$$

Steps Using Factoring By Grouping

Subtract 18r from both sides.

$$-4r^{2}-18r=18$$

Subtract 18 from both sides.

$$-4r^{2}-18r-18=0$$

Divide both sides by 2.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2r^{2}+ar+br-9. To find a and b, set up a system to be solved.

$$a+b=-9 ab=-2\left(-9\right)=18$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -9.

$$a=-3 b=-6$$

Rewrite -2r^{2}-9r-9 as \left(-2r^{2}-3r\right)+\left(-6r-9\right).

$$\left(-2r^{2}-3r\right)+\left(-6r-9\right)$$

Factor out -r in the first and -3 in the second group.

$$-r\left(2r+3\right)-3\left(2r+3\right)$$

Factor out common term 2r+3 by using distributive property.

$$\left(2r+3\right)\left(-r-3\right)$$

To find equation solutions, solve 2r+3=0 and -r-3=0.

$$r=-\frac{3}{2} r=-3$$

Steps Using the Quadratic Formula

Subtract 18r from both sides.

$$-4r^{2}-18r=18$$

Subtract 18 from both sides.

$$-4r^{2}-18r-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -18 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-4\right)\left(-18\right)}}{2\left(-4\right)}$$

Square -18.

$$r=\frac{-\left(-18\right)±\sqrt{324-4\left(-4\right)\left(-18\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$r=\frac{-\left(-18\right)±\sqrt{324+16\left(-18\right)}}{2\left(-4\right)}$$

Multiply 16 times -18.

$$r=\frac{-\left(-18\right)±\sqrt{324-288}}{2\left(-4\right)}$$

Add 324 to -288.

$$r=\frac{-\left(-18\right)±\sqrt{36}}{2\left(-4\right)}$$

Take the square root of 36.

$$r=\frac{-\left(-18\right)±6}{2\left(-4\right)}$$

The opposite of -18 is 18.

$$r=\frac{18±6}{2\left(-4\right)}$$

Multiply 2 times -4.

$$r=\frac{18±6}{-8}$$

Now solve the equation r=\frac{18±6}{-8} when ± is plus. Add 18 to 6.

$$r=\frac{24}{-8}$$

Divide 24 by -8.

$$r=-3$$

Now solve the equation r=\frac{18±6}{-8} when ± is minus. Subtract 6 from 18.

$$r=\frac{12}{-8}$$

Reduce the fraction \frac{12}{-8} to lowest terms by extracting and canceling out 4.

$$r=-\frac{3}{2}$$

The equation is now solved.

$$r=-3 r=-\frac{3}{2}$$

Steps for Completing the Square

Subtract 18r from both sides.

$$-4r^{2}-18r=18$$

Divide both sides by -4.

$$\frac{-4r^{2}-18r}{-4}=\frac{18}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$r^{2}+\left(-\frac{18}{-4}\right)r=\frac{18}{-4}$$

Reduce the fraction \frac{-18}{-4} to lowest terms by extracting and canceling out 2.

$$r^{2}+\frac{9}{2}r=\frac{18}{-4}$$

Reduce the fraction \frac{18}{-4} to lowest terms by extracting and canceling out 2.

$$r^{2}+\frac{9}{2}r=-\frac{9}{2}$$

Divide \frac{9}{2}, the coefficient of the x term, by 2 to get \frac{9}{4}. Then add the square of \frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+\frac{9}{2}r+\left(\frac{9}{4}\right)^{2}=-\frac{9}{2}+\left(\frac{9}{4}\right)^{2}$$

Square \frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$r^{2}+\frac{9}{2}r+\frac{81}{16}=-\frac{9}{2}+\frac{81}{16}$$

Add -\frac{9}{2} to \frac{81}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$r^{2}+\frac{9}{2}r+\frac{81}{16}=\frac{9}{16}$$

Factor r^{2}+\frac{9}{2}r+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+\frac{9}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+\frac{9}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

$$r+\frac{9}{4}=\frac{3}{4} r+\frac{9}{4}=-\frac{3}{4}$$

Subtract \frac{9}{4} from both sides of the equation.

$$r=-\frac{3}{2} r=-3$$