Solve (y-2)^2=25

$$( y - 2 ) ^ { 2 } = 25$$

Solve for y

$$y=7$$

$$y=-3$$

Steps Using Factoring

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

$$y^{2}-4y+4=25$$

Subtract 25 from both sides.

$$y^{2}-4y+4-25=0$$

Subtract 25 from 4 to get -21.

$$y^{2}-4y-21=0$$

To solve the equation, factor y^{2}-4y-21 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.

$$a+b=-4 ab=-21$$

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. List all such integer pairs that give product -21.

$$1,-21 3,-7$$

Calculate the sum for each pair.

$$1-21=-20 3-7=-4$$

The solution is the pair that gives sum -4.

$$a=-7 b=3$$

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

$$\left(y-7\right)\left(y+3\right)$$

To find equation solutions, solve y-7=0 and y+3=0.

$$y=7 y=-3$$

Steps Using Factoring By Grouping

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

$$y^{2}-4y+4=25$$

Subtract 25 from both sides.

$$y^{2}-4y+4-25=0$$

Subtract 25 from 4 to get -21.

$$y^{2}-4y-21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as y^{2}+ay+by-21. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-21\right)=-21$$

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. List all such integer pairs that give product -21.

$$1,-21 3,-7$$

Calculate the sum for each pair.

$$1-21=-20 3-7=-4$$

The solution is the pair that gives sum -4.

$$a=-7 b=3$$

Rewrite y^{2}-4y-21 as \left(y^{2}-7y\right)+\left(3y-21\right).

$$\left(y^{2}-7y\right)+\left(3y-21\right)$$

Factor out y in the first and 3 in the second group.

$$y\left(y-7\right)+3\left(y-7\right)$$

Factor out common term y-7 by using distributive property.

$$\left(y-7\right)\left(y+3\right)$$

To find equation solutions, solve y-7=0 and y+3=0.

$$y=7 y=-3$$

Steps Using the Quadratic Formula

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

$$y^{2}-4y+4=25$$

Subtract 25 from both sides.

$$y^{2}-4y+4-25=0$$

Subtract 25 from 4 to get -21.

$$y^{2}-4y-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-21\right)}}{2}$$

Square -4.

$$y=\frac{-\left(-4\right)±\sqrt{16-4\left(-21\right)}}{2}$$

Multiply -4 times -21.

$$y=\frac{-\left(-4\right)±\sqrt{16+84}}{2}$$

Add 16 to 84.

$$y=\frac{-\left(-4\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$y=\frac{-\left(-4\right)±10}{2}$$

The opposite of -4 is 4.

$$y=\frac{4±10}{2}$$

Now solve the equation y=\frac{4±10}{2} when ± is plus. Add 4 to 10.

$$y=\frac{14}{2}$$

Divide 14 by 2.

$$y=7$$

Now solve the equation y=\frac{4±10}{2} when ± is minus. Subtract 10 from 4.

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

Divide -6 by 2.

$$y=-3$$

The equation is now solved.

$$y=7 y=-3$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(y-2\right)^{2}}=\sqrt{25}$$

Simplify.

$$y-2=5 y-2=-5$$

Add 2 to both sides of the equation.

$$y=7 y=-3$$