Solve -2(y+7)^2=-128

$$- 2 ( y + 7 ) ^ { 2 } = - 128$$

Solve for y

$$y=1$$

$$y=-15$$

Steps Using Factoring

Divide both sides by -2.

$$\left(y+7\right)^{2}=\frac{-128}{-2}$$

Divide -128 by -2 to get 64.

$$\left(y+7\right)^{2}=64$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+7\right)^{2}.

$$y^{2}+14y+49=64$$

Subtract 64 from both sides.

$$y^{2}+14y+49-64=0$$

Subtract 64 from 49 to get -15.

$$y^{2}+14y-15=0$$

To solve the equation, factor y^{2}+14y-15 using formula y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). To find a and b, set up a system to be solved.

$$a+b=14 ab=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 14.

$$a=-1 b=15$$

Rewrite factored expression \left(y+a\right)\left(y+b\right) using the obtained values.

$$\left(y-1\right)\left(y+15\right)$$

To find equation solutions, solve y-1=0 and y+15=0.

$$y=1 y=-15$$

Steps Using Factoring By Grouping

Divide both sides by -2.

$$\left(y+7\right)^{2}=\frac{-128}{-2}$$

Divide -128 by -2 to get 64.

$$\left(y+7\right)^{2}=64$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+7\right)^{2}.

$$y^{2}+14y+49=64$$

Subtract 64 from both sides.

$$y^{2}+14y+49-64=0$$

Subtract 64 from 49 to get -15.

$$y^{2}+14y-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as y^{2}+ay+by-15. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\left(-15\right)=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 14.

$$a=-1 b=15$$

Rewrite y^{2}+14y-15 as \left(y^{2}-y\right)+\left(15y-15\right).

$$\left(y^{2}-y\right)+\left(15y-15\right)$$

Factor out y in the first and 15 in the second group.

$$y\left(y-1\right)+15\left(y-1\right)$$

Factor out common term y-1 by using distributive property.

$$\left(y-1\right)\left(y+15\right)$$

To find equation solutions, solve y-1=0 and y+15=0.

$$y=1 y=-15$$

Steps Using the Quadratic Formula

Divide both sides by -2.

$$\left(y+7\right)^{2}=\frac{-128}{-2}$$

Divide -128 by -2 to get 64.

$$\left(y+7\right)^{2}=64$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+7\right)^{2}.

$$y^{2}+14y+49=64$$

Subtract 64 from both sides.

$$y^{2}+14y+49-64=0$$

Subtract 64 from 49 to get -15.

$$y^{2}+14y-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-14±\sqrt{14^{2}-4\left(-15\right)}}{2}$$

Square 14.

$$y=\frac{-14±\sqrt{196-4\left(-15\right)}}{2}$$

Multiply -4 times -15.

$$y=\frac{-14±\sqrt{196+60}}{2}$$

Add 196 to 60.

$$y=\frac{-14±\sqrt{256}}{2}$$

Take the square root of 256.

$$y=\frac{-14±16}{2}$$

Now solve the equation y=\frac{-14±16}{2} when ± is plus. Add -14 to 16.

$$y=\frac{2}{2}$$

Divide 2 by 2.

$$y=1$$

Now solve the equation y=\frac{-14±16}{2} when ± is minus. Subtract 16 from -14.

$$y=-\frac{30}{2}$$

Divide -30 by 2.

$$y=-15$$

The equation is now solved.

$$y=1 y=-15$$

Steps for Completing the Square

Divide both sides by -2.

$$\left(y+7\right)^{2}=\frac{-128}{-2}$$

Divide -128 by -2 to get 64.

$$\left(y+7\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+7\right)^{2}}=\sqrt{64}$$

Simplify.

$$y+7=8 y+7=-8$$

Subtract 7 from both sides of the equation.

$$y=1 y=-15$$