Solve l=l^2+3/16

$$l = l ^ { 2 } + \frac { 3 } { 16 }$$

Solve for l

$$l=\frac{1}{4}=0.25$$

$$l=\frac{3}{4}=0.75$$

Steps Using the Quadratic Formula

Subtract l^{2} from both sides.

$$l-l^{2}=\frac{3}{16}$$

Subtract \frac{3}{16} from both sides.

$$l-l^{2}-\frac{3}{16}=0$$

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.

$$-l^{2}+l-\frac{3}{16}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and -\frac{3}{16} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$l=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\left(-\frac{3}{16}\right)}}{2\left(-1\right)}$$

Square 1.

$$l=\frac{-1±\sqrt{1-4\left(-1\right)\left(-\frac{3}{16}\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$l=\frac{-1±\sqrt{1+4\left(-\frac{3}{16}\right)}}{2\left(-1\right)}$$

Multiply 4 times -\frac{3}{16}.

$$l=\frac{-1±\sqrt{1-\frac{3}{4}}}{2\left(-1\right)}$$

Add 1 to -\frac{3}{4}.

$$l=\frac{-1±\sqrt{\frac{1}{4}}}{2\left(-1\right)}$$

Take the square root of \frac{1}{4}.

$$l=\frac{-1±\frac{1}{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$l=\frac{-1±\frac{1}{2}}{-2}$$

Now solve the equation l=\frac{-1±\frac{1}{2}}{-2} when ± is plus. Add -1 to \frac{1}{2}.

$$l=-\frac{\frac{1}{2}}{-2}$$

Divide -\frac{1}{2} by -2.

$$l=\frac{1}{4}$$

Now solve the equation l=\frac{-1±\frac{1}{2}}{-2} when ± is minus. Subtract \frac{1}{2} from -1.

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

Divide -\frac{3}{2} by -2.

$$l=\frac{3}{4}$$

The equation is now solved.

$$l=\frac{1}{4} l=\frac{3}{4}$$

Steps for Completing the Square

Subtract l^{2} from both sides.

$$l-l^{2}=\frac{3}{16}$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-l^{2}+l=\frac{3}{16}$$

Divide both sides by -1.

$$\frac{-l^{2}+l}{-1}=\frac{\frac{3}{16}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$l^{2}+\frac{1}{-1}l=\frac{\frac{3}{16}}{-1}$$

Divide 1 by -1.

$$l^{2}-l=\frac{\frac{3}{16}}{-1}$$

Divide \frac{3}{16} by -1.

$$l^{2}-l=-\frac{3}{16}$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$l^{2}-l+\left(-\frac{1}{2}\right)^{2}=-\frac{3}{16}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$l^{2}-l+\frac{1}{4}=-\frac{3}{16}+\frac{1}{4}$$

Add -\frac{3}{16} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$l^{2}-l+\frac{1}{4}=\frac{1}{16}$$

Factor l^{2}-l+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(l-\frac{1}{2}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(l-\frac{1}{2}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$l-\frac{1}{2}=\frac{1}{4} l-\frac{1}{2}=-\frac{1}{4}$$

Add \frac{1}{2} to both sides of the equation.

$$l=\frac{3}{4} l=\frac{1}{4}$$