Solve (a-4)^2=25

$$( a - 4 ) ^ { 2 } = 25$$

Solve for a

$$a=9$$

$$a=-1$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-4\right)^{2}.

$$a^{2}-8a+16=25$$

Subtract 25 from both sides.

$$a^{2}-8a+16-25=0$$

Subtract 25 from 16 to get -9.

$$a^{2}-8a-9=0$$

To solve the equation, factor a^{2}-8a-9 using formula a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+b\right). To find a and b, set up a system to be solved.

$$a+b=-8 ab=-9$$

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 -9.

$$1,-9 3,-3$$

Calculate the sum for each pair.

$$1-9=-8 3-3=0$$

The solution is the pair that gives sum -8.

$$a=-9 b=1$$

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

$$\left(a-9\right)\left(a+1\right)$$

To find equation solutions, solve a-9=0 and a+1=0.

$$a=9 a=-1$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-4\right)^{2}.

$$a^{2}-8a+16=25$$

Subtract 25 from both sides.

$$a^{2}-8a+16-25=0$$

Subtract 25 from 16 to get -9.

$$a^{2}-8a-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as a^{2}+aa+ba-9. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\left(-9\right)=-9$$

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 -9.

$$1,-9 3,-3$$

Calculate the sum for each pair.

$$1-9=-8 3-3=0$$

The solution is the pair that gives sum -8.

$$a=-9 b=1$$

Rewrite a^{2}-8a-9 as \left(a^{2}-9a\right)+\left(a-9\right).

$$\left(a^{2}-9a\right)+\left(a-9\right)$$

Factor out a in a^{2}-9a.

$$a\left(a-9\right)+a-9$$

Factor out common term a-9 by using distributive property.

$$\left(a-9\right)\left(a+1\right)$$

To find equation solutions, solve a-9=0 and a+1=0.

$$a=9 a=-1$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-4\right)^{2}.

$$a^{2}-8a+16=25$$

Subtract 25 from both sides.

$$a^{2}-8a+16-25=0$$

Subtract 25 from 16 to get -9.

$$a^{2}-8a-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-9\right)}}{2}$$

Square -8.

$$a=\frac{-\left(-8\right)±\sqrt{64-4\left(-9\right)}}{2}$$

Multiply -4 times -9.

$$a=\frac{-\left(-8\right)±\sqrt{64+36}}{2}$$

Add 64 to 36.

$$a=\frac{-\left(-8\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$a=\frac{-\left(-8\right)±10}{2}$$

The opposite of -8 is 8.

$$a=\frac{8±10}{2}$$

Now solve the equation a=\frac{8±10}{2} when ± is plus. Add 8 to 10.

$$a=\frac{18}{2}$$

Divide 18 by 2.

$$a=9$$

Now solve the equation a=\frac{8±10}{2} when ± is minus. Subtract 10 from 8.

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

Divide -2 by 2.

$$a=-1$$

The equation is now solved.

$$a=9 a=-1$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(a-4\right)^{2}}=\sqrt{25}$$

Simplify.

$$a-4=5 a-4=-5$$

Add 4 to both sides of the equation.

$$a=9 a=-1$$