Solve y^2+10=-12y

$$y ^ { 2 } + 10 = - 12 y$$

Solve for y (complex solution)

$$y=\sqrt{26}-6\approx -0.900980486$$

$$y=-\left(\sqrt{26}+6\right)\approx -11.099019514$$

Steps Using the Quadratic Formula

Add 12y to both sides.

$$y^{2}+10+12y=0$$

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.

$$y^{2}+12y+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 12 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-12±\sqrt{12^{2}-4\times 10}}{2}$$

Square 12.

$$y=\frac{-12±\sqrt{144-4\times 10}}{2}$$

Multiply -4 times 10.

$$y=\frac{-12±\sqrt{144-40}}{2}$$

Add 144 to -40.

$$y=\frac{-12±\sqrt{104}}{2}$$

Take the square root of 104.

$$y=\frac{-12±2\sqrt{26}}{2}$$

Now solve the equation y=\frac{-12±2\sqrt{26}}{2} when ± is plus. Add -12 to 2\sqrt{26}.

$$y=\frac{2\sqrt{26}-12}{2}$$

Divide -12+2\sqrt{26} by 2.

$$y=\sqrt{26}-6$$

Now solve the equation y=\frac{-12±2\sqrt{26}}{2} when ± is minus. Subtract 2\sqrt{26} from -12.

$$y=\frac{-2\sqrt{26}-12}{2}$$

Divide -12-2\sqrt{26} by 2.

$$y=-\sqrt{26}-6$$

The equation is now solved.

$$y=\sqrt{26}-6 y=-\sqrt{26}-6$$

Steps for Completing the Square

Add 12y to both sides.

$$y^{2}+10+12y=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$y^{2}+12y=-10$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+12y+6^{2}=-10+6^{2}$$

Square 6.

$$y^{2}+12y+36=-10+36$$

Add -10 to 36.

$$y^{2}+12y+36=26$$

Factor y^{2}+12y+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+6\right)^{2}=26$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+6\right)^{2}}=\sqrt{26}$$

Simplify.

$$y+6=\sqrt{26} y+6=-\sqrt{26}$$

Subtract 6 from both sides of the equation.

$$y=\sqrt{26}-6 y=-\sqrt{26}-6$$

Solve for y

$$y=\sqrt{26}-6\approx -0.900980486$$

$$y=-\sqrt{26}-6\approx -11.099019514$$

Steps Using the Quadratic Formula

Add 12y to both sides.

$$y^{2}+10+12y=0$$

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.

$$y^{2}+12y+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 12 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-12±\sqrt{12^{2}-4\times 10}}{2}$$

Square 12.

$$y=\frac{-12±\sqrt{144-4\times 10}}{2}$$

Multiply -4 times 10.

$$y=\frac{-12±\sqrt{144-40}}{2}$$

Add 144 to -40.

$$y=\frac{-12±\sqrt{104}}{2}$$

Take the square root of 104.

$$y=\frac{-12±2\sqrt{26}}{2}$$

Now solve the equation y=\frac{-12±2\sqrt{26}}{2} when ± is plus. Add -12 to 2\sqrt{26}.

$$y=\frac{2\sqrt{26}-12}{2}$$

Divide -12+2\sqrt{26} by 2.

$$y=\sqrt{26}-6$$

Now solve the equation y=\frac{-12±2\sqrt{26}}{2} when ± is minus. Subtract 2\sqrt{26} from -12.

$$y=\frac{-2\sqrt{26}-12}{2}$$

Divide -12-2\sqrt{26} by 2.

$$y=-\sqrt{26}-6$$

The equation is now solved.

$$y=\sqrt{26}-6 y=-\sqrt{26}-6$$

Steps for Completing the Square

Add 12y to both sides.

$$y^{2}+10+12y=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$y^{2}+12y=-10$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+12y+6^{2}=-10+6^{2}$$

Square 6.

$$y^{2}+12y+36=-10+36$$

Add -10 to 36.

$$y^{2}+12y+36=26$$

Factor y^{2}+12y+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+6\right)^{2}=26$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+6\right)^{2}}=\sqrt{26}$$

Simplify.

$$y+6=\sqrt{26} y+6=-\sqrt{26}$$

Subtract 6 from both sides of the equation.

$$y=\sqrt{26}-6 y=-\sqrt{26}-6$$