Solve y^2-16y+64=81

$$y ^ { 2 } - 16 y + 64 = 81$$

Solve for y

$$y=-1$$

$$y=17$$

Steps Using Factoring

Subtract 81 from both sides.

$$y^{2}-16y+64-81=0$$

Subtract 81 from 64 to get -17.

$$y^{2}-16y-17=0$$

To solve the equation, factor y^{2}-16y-17 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.

$$a+b=-16 ab=-17$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-17 b=1$$

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

$$\left(y-17\right)\left(y+1\right)$$

To find equation solutions, solve y-17=0 and y+1=0.

$$y=17 y=-1$$

Steps Using Factoring By Grouping

Subtract 81 from both sides.

$$y^{2}-16y+64-81=0$$

Subtract 81 from 64 to get -17.

$$y^{2}-16y-17=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as y^{2}+ay+by-17. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\left(-17\right)=-17$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-17 b=1$$

Rewrite y^{2}-16y-17 as \left(y^{2}-17y\right)+\left(y-17\right).

$$\left(y^{2}-17y\right)+\left(y-17\right)$$

Factor out y in y^{2}-17y.

$$y\left(y-17\right)+y-17$$

Factor out common term y-17 by using distributive property.

$$\left(y-17\right)\left(y+1\right)$$

To find equation solutions, solve y-17=0 and y+1=0.

$$y=17 y=-1$$

Steps Using the Quadratic Formula

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.

$$y^{2}-16y+64=81$$

Subtract 81 from both sides of the equation.

$$y^{2}-16y+64-81=81-81$$

Subtracting 81 from itself leaves 0.

$$y^{2}-16y+64-81=0$$

Subtract 81 from 64.

$$y^{2}-16y-17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and -17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-17\right)}}{2}$$

Square -16.

$$y=\frac{-\left(-16\right)±\sqrt{256-4\left(-17\right)}}{2}$$

Multiply -4 times -17.

$$y=\frac{-\left(-16\right)±\sqrt{256+68}}{2}$$

Add 256 to 68.

$$y=\frac{-\left(-16\right)±\sqrt{324}}{2}$$

Take the square root of 324.

$$y=\frac{-\left(-16\right)±18}{2}$$

The opposite of -16 is 16.

$$y=\frac{16±18}{2}$$

Now solve the equation y=\frac{16±18}{2} when ± is plus. Add 16 to 18.

$$y=\frac{34}{2}$$

Divide 34 by 2.

$$y=17$$

Now solve the equation y=\frac{16±18}{2} when ± is minus. Subtract 18 from 16.

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

Divide -2 by 2.

$$y=-1$$

The equation is now solved.

$$y=17 y=-1$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$y^{2}-16y+64=81$$

Factor y^{2}-16y+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-8\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-8\right)^{2}}=\sqrt{81}$$

Simplify.

$$y-8=9 y-8=-9$$

Add 8 to both sides of the equation.

$$y=17 y=-1$$