Solve 0=y^2+6y-14

$$0 = y ^ { 2 } + 6 y - 14$$

Solve for y (complex solution)

$$y=\sqrt{23}-3\approx 1.795831523$$

$$y=-\left(\sqrt{23}+3\right)\approx -7.795831523$$

Steps Using the Quadratic Formula

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

$$y^{2}+6y-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-6±\sqrt{6^{2}-4\left(-14\right)}}{2}$$

Square 6.

$$y=\frac{-6±\sqrt{36-4\left(-14\right)}}{2}$$

Multiply -4 times -14.

$$y=\frac{-6±\sqrt{36+56}}{2}$$

Add 36 to 56.

$$y=\frac{-6±\sqrt{92}}{2}$$

Take the square root of 92.

$$y=\frac{-6±2\sqrt{23}}{2}$$

Now solve the equation y=\frac{-6±2\sqrt{23}}{2} when ± is plus. Add -6 to 2\sqrt{23}.

$$y=\frac{2\sqrt{23}-6}{2}$$

Divide -6+2\sqrt{23} by 2.

$$y=\sqrt{23}-3$$

Now solve the equation y=\frac{-6±2\sqrt{23}}{2} when ± is minus. Subtract 2\sqrt{23} from -6.

$$y=\frac{-2\sqrt{23}-6}{2}$$

Divide -6-2\sqrt{23} by 2.

$$y=-\sqrt{23}-3$$

The equation is now solved.

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$

Steps for Completing the Square

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

$$y^{2}+6y-14=0$$

Add 14 to both sides. Anything plus zero gives itself.

$$y^{2}+6y=14$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+6y+3^{2}=14+3^{2}$$

Square 3.

$$y^{2}+6y+9=14+9$$

Add 14 to 9.

$$y^{2}+6y+9=23$$

Factor y^{2}+6y+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+3\right)^{2}=23$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{23}$$

Simplify.

$$y+3=\sqrt{23} y+3=-\sqrt{23}$$

Subtract 3 from both sides of the equation.

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$

Solve for y

$$y=\sqrt{23}-3\approx 1.795831523$$

$$y=-\sqrt{23}-3\approx -7.795831523$$

Steps Using the Quadratic Formula

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

$$y^{2}+6y-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-6±\sqrt{6^{2}-4\left(-14\right)}}{2}$$

Square 6.

$$y=\frac{-6±\sqrt{36-4\left(-14\right)}}{2}$$

Multiply -4 times -14.

$$y=\frac{-6±\sqrt{36+56}}{2}$$

Add 36 to 56.

$$y=\frac{-6±\sqrt{92}}{2}$$

Take the square root of 92.

$$y=\frac{-6±2\sqrt{23}}{2}$$

Now solve the equation y=\frac{-6±2\sqrt{23}}{2} when ± is plus. Add -6 to 2\sqrt{23}.

$$y=\frac{2\sqrt{23}-6}{2}$$

Divide -6+2\sqrt{23} by 2.

$$y=\sqrt{23}-3$$

Now solve the equation y=\frac{-6±2\sqrt{23}}{2} when ± is minus. Subtract 2\sqrt{23} from -6.

$$y=\frac{-2\sqrt{23}-6}{2}$$

Divide -6-2\sqrt{23} by 2.

$$y=-\sqrt{23}-3$$

The equation is now solved.

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$

Steps for Completing the Square

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

$$y^{2}+6y-14=0$$

Add 14 to both sides. Anything plus zero gives itself.

$$y^{2}+6y=14$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+6y+3^{2}=14+3^{2}$$

Square 3.

$$y^{2}+6y+9=14+9$$

Add 14 to 9.

$$y^{2}+6y+9=23$$

Factor y^{2}+6y+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+3\right)^{2}=23$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{23}$$

Simplify.

$$y+3=\sqrt{23} y+3=-\sqrt{23}$$

Subtract 3 from both sides of the equation.

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$