Solve y^2=21+2y/2

$$y ^ { 2 } = \frac { 21 + 2 y } { 2 }$$

Solve for y

$$y = \frac{\sqrt{43} + 1}{2} \approx 3.778719262$$

$$y=\frac{1-\sqrt{43}}{2}\approx -2.778719262$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2y^{2}=21+2y$$

Subtract 21 from both sides.

$$2y^{2}-21=2y$$

Subtract 2y from both sides.

$$2y^{2}-21-2y=0$$

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.

$$2y^{2}-2y-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -2 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-21\right)}}{2\times 2}$$

Square -2.

$$y=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-21\right)}}{2\times 2}$$

Multiply -4 times 2.

$$y=\frac{-\left(-2\right)±\sqrt{4-8\left(-21\right)}}{2\times 2}$$

Multiply -8 times -21.

$$y=\frac{-\left(-2\right)±\sqrt{4+168}}{2\times 2}$$

Add 4 to 168.

$$y=\frac{-\left(-2\right)±\sqrt{172}}{2\times 2}$$

Take the square root of 172.

$$y=\frac{-\left(-2\right)±2\sqrt{43}}{2\times 2}$$

The opposite of -2 is 2.

$$y=\frac{2±2\sqrt{43}}{2\times 2}$$

Multiply 2 times 2.

$$y=\frac{2±2\sqrt{43}}{4}$$

Now solve the equation y=\frac{2±2\sqrt{43}}{4} when ± is plus. Add 2 to 2\sqrt{43}.

$$y=\frac{2\sqrt{43}+2}{4}$$

Divide 2+2\sqrt{43} by 4.

$$y=\frac{\sqrt{43}+1}{2}$$

Now solve the equation y=\frac{2±2\sqrt{43}}{4} when ± is minus. Subtract 2\sqrt{43} from 2.

$$y=\frac{2-2\sqrt{43}}{4}$$

Divide 2-2\sqrt{43} by 4.

$$y=\frac{1-\sqrt{43}}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{43}+1}{2} y=\frac{1-\sqrt{43}}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2y^{2}=21+2y$$

Subtract 2y from both sides.

$$2y^{2}-2y=21$$

Divide both sides by 2.

$$\frac{2y^{2}-2y}{2}=\frac{21}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$y^{2}+\left(-\frac{2}{2}\right)y=\frac{21}{2}$$

Divide -2 by 2.

$$y^{2}-y=\frac{21}{2}$$

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.

$$y^{2}-y+\left(-\frac{1}{2}\right)^{2}=\frac{21}{2}+\left(-\frac{1}{2}\right)^{2}$$

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

$$y^{2}-y+\frac{1}{4}=\frac{21}{2}+\frac{1}{4}$$

Add \frac{21}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-y+\frac{1}{4}=\frac{43}{4}$$

Factor y^{2}-y+\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}.

$$\left(y-\frac{1}{2}\right)^{2}=\frac{43}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{1}{2}\right)^{2}}=\sqrt{\frac{43}{4}}$$

Simplify.

$$y-\frac{1}{2}=\frac{\sqrt{43}}{2} y-\frac{1}{2}=-\frac{\sqrt{43}}{2}$$

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

$$y=\frac{\sqrt{43}+1}{2} y=\frac{1-\sqrt{43}}{2}$$