Solve 5=4r(2r+3)

$$5 = 4 r ( 2 r + 3 )$$

Solve for r

$$r=\frac{\sqrt{19}-3}{4}\approx 0.339724736$$

$$r=\frac{-\sqrt{19}-3}{4}\approx -1.839724736$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4r by 2r+3.

$$5=8r^{2}+12r$$

Swap sides so that all variable terms are on the left hand side.

$$8r^{2}+12r=5$$

Subtract 5 from both sides.

$$8r^{2}+12r-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 12 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-12±\sqrt{12^{2}-4\times 8\left(-5\right)}}{2\times 8}$$

Square 12.

$$r=\frac{-12±\sqrt{144-4\times 8\left(-5\right)}}{2\times 8}$$

Multiply -4 times 8.

$$r=\frac{-12±\sqrt{144-32\left(-5\right)}}{2\times 8}$$

Multiply -32 times -5.

$$r=\frac{-12±\sqrt{144+160}}{2\times 8}$$

Add 144 to 160.

$$r=\frac{-12±\sqrt{304}}{2\times 8}$$

Take the square root of 304.

$$r=\frac{-12±4\sqrt{19}}{2\times 8}$$

Multiply 2 times 8.

$$r=\frac{-12±4\sqrt{19}}{16}$$

Now solve the equation r=\frac{-12±4\sqrt{19}}{16} when ± is plus. Add -12 to 4\sqrt{19}.

$$r=\frac{4\sqrt{19}-12}{16}$$

Divide -12+4\sqrt{19} by 16.

$$r=\frac{\sqrt{19}-3}{4}$$

Now solve the equation r=\frac{-12±4\sqrt{19}}{16} when ± is minus. Subtract 4\sqrt{19} from -12.

$$r=\frac{-4\sqrt{19}-12}{16}$$

Divide -12-4\sqrt{19} by 16.

$$r=\frac{-\sqrt{19}-3}{4}$$

The equation is now solved.

$$r=\frac{\sqrt{19}-3}{4} r=\frac{-\sqrt{19}-3}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 4r by 2r+3.

$$5=8r^{2}+12r$$

Swap sides so that all variable terms are on the left hand side.

$$8r^{2}+12r=5$$

Divide both sides by 8.

$$\frac{8r^{2}+12r}{8}=\frac{5}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$r^{2}+\frac{12}{8}r=\frac{5}{8}$$

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

$$r^{2}+\frac{3}{2}r=\frac{5}{8}$$

Divide \frac{3}{2}, the coefficient of the x term, by 2 to get \frac{3}{4}. Then add the square of \frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+\frac{3}{2}r+\left(\frac{3}{4}\right)^{2}=\frac{5}{8}+\left(\frac{3}{4}\right)^{2}$$

Square \frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$r^{2}+\frac{3}{2}r+\frac{9}{16}=\frac{5}{8}+\frac{9}{16}$$

Add \frac{5}{8} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$r^{2}+\frac{3}{2}r+\frac{9}{16}=\frac{19}{16}$$

Factor r^{2}+\frac{3}{2}r+\frac{9}{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}.

$$\left(r+\frac{3}{4}\right)^{2}=\frac{19}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+\frac{3}{4}\right)^{2}}=\sqrt{\frac{19}{16}}$$

Simplify.

$$r+\frac{3}{4}=\frac{\sqrt{19}}{4} r+\frac{3}{4}=-\frac{\sqrt{19}}{4}$$

Subtract \frac{3}{4} from both sides of the equation.

$$r=\frac{\sqrt{19}-3}{4} r=\frac{-\sqrt{19}-3}{4}$$