Solve 594=20l^2+101l

$$594 = 20 l ^ { 2 } + 101 l$$

Solve for l

$$l = \frac{\sqrt{57721} - 101}{40} \approx 3.481298777$$

$$l=\frac{-\sqrt{57721}-101}{40}\approx -8.531298777$$

Steps Using the Quadratic Formula

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

$$20l^{2}+101l=594$$

Subtract 594 from both sides.

$$20l^{2}+101l-594=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, 101 for b, and -594 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$l=\frac{-101±\sqrt{101^{2}-4\times 20\left(-594\right)}}{2\times 20}$$

Square 101.

$$l=\frac{-101±\sqrt{10201-4\times 20\left(-594\right)}}{2\times 20}$$

Multiply -4 times 20.

$$l=\frac{-101±\sqrt{10201-80\left(-594\right)}}{2\times 20}$$

Multiply -80 times -594.

$$l=\frac{-101±\sqrt{10201+47520}}{2\times 20}$$

Add 10201 to 47520.

$$l=\frac{-101±\sqrt{57721}}{2\times 20}$$

Multiply 2 times 20.

$$l=\frac{-101±\sqrt{57721}}{40}$$

Now solve the equation l=\frac{-101±\sqrt{57721}}{40} when ± is plus. Add -101 to \sqrt{57721}.

$$l=\frac{\sqrt{57721}-101}{40}$$

Now solve the equation l=\frac{-101±\sqrt{57721}}{40} when ± is minus. Subtract \sqrt{57721} from -101.

$$l=\frac{-\sqrt{57721}-101}{40}$$

The equation is now solved.

$$l=\frac{\sqrt{57721}-101}{40} l=\frac{-\sqrt{57721}-101}{40}$$

Steps for Completing the Square

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

$$20l^{2}+101l=594$$

Divide both sides by 20.

$$\frac{20l^{2}+101l}{20}=\frac{594}{20}$$

Dividing by 20 undoes the multiplication by 20.

$$l^{2}+\frac{101}{20}l=\frac{594}{20}$$

Reduce the fraction \frac{594}{20} to lowest terms by extracting and canceling out 2.

$$l^{2}+\frac{101}{20}l=\frac{297}{10}$$

Divide \frac{101}{20}, the coefficient of the x term, by 2 to get \frac{101}{40}. Then add the square of \frac{101}{40} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$l^{2}+\frac{101}{20}l+\left(\frac{101}{40}\right)^{2}=\frac{297}{10}+\left(\frac{101}{40}\right)^{2}$$

Square \frac{101}{40} by squaring both the numerator and the denominator of the fraction.

$$l^{2}+\frac{101}{20}l+\frac{10201}{1600}=\frac{297}{10}+\frac{10201}{1600}$$

Add \frac{297}{10} to \frac{10201}{1600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$l^{2}+\frac{101}{20}l+\frac{10201}{1600}=\frac{57721}{1600}$$

Factor l^{2}+\frac{101}{20}l+\frac{10201}{1600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(l+\frac{101}{40}\right)^{2}=\frac{57721}{1600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(l+\frac{101}{40}\right)^{2}}=\sqrt{\frac{57721}{1600}}$$

Simplify.

$$l+\frac{101}{40}=\frac{\sqrt{57721}}{40} l+\frac{101}{40}=-\frac{\sqrt{57721}}{40}$$

Subtract \frac{101}{40} from both sides of the equation.

$$l=\frac{\sqrt{57721}-101}{40} l=\frac{-\sqrt{57721}-101}{40}$$