Solve y^2=9y+14

$$y ^ { 2 } = 9 y + 14$$

Solve for y

$$y = \frac{\sqrt{137} + 9}{2} \approx 10.352349955$$

$$y=\frac{9-\sqrt{137}}{2}\approx -1.352349955$$

Steps Using the Quadratic Formula

Subtract 9y from both sides.

$$y^{2}-9y=14$$

Subtract 14 from both sides.

$$y^{2}-9y-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-14\right)}}{2}$$

Square -9.

$$y=\frac{-\left(-9\right)±\sqrt{81-4\left(-14\right)}}{2}$$

Multiply -4 times -14.

$$y=\frac{-\left(-9\right)±\sqrt{81+56}}{2}$$

Add 81 to 56.

$$y=\frac{-\left(-9\right)±\sqrt{137}}{2}$$

The opposite of -9 is 9.

$$y=\frac{9±\sqrt{137}}{2}$$

Now solve the equation y=\frac{9±\sqrt{137}}{2} when ± is plus. Add 9 to \sqrt{137}.

$$y=\frac{\sqrt{137}+9}{2}$$

Now solve the equation y=\frac{9±\sqrt{137}}{2} when ± is minus. Subtract \sqrt{137} from 9.

$$y=\frac{9-\sqrt{137}}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{137}+9}{2} y=\frac{9-\sqrt{137}}{2}$$

Steps for Completing the Square

Subtract 9y from both sides.

$$y^{2}-9y=14$$

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

$$y^{2}-9y+\left(-\frac{9}{2}\right)^{2}=14+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-9y+\frac{81}{4}=14+\frac{81}{4}$$

Add 14 to \frac{81}{4}.

$$y^{2}-9y+\frac{81}{4}=\frac{137}{4}$$

Factor y^{2}-9y+\frac{81}{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}.

$$\left(y-\frac{9}{2}\right)^{2}=\frac{137}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{9}{2}\right)^{2}}=\sqrt{\frac{137}{4}}$$

Simplify.

$$y-\frac{9}{2}=\frac{\sqrt{137}}{2} y-\frac{9}{2}=-\frac{\sqrt{137}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$y=\frac{\sqrt{137}+9}{2} y=\frac{9-\sqrt{137}}{2}$$