Solve (p-4)^2=49

$$( p - 4 ) ^ { 2 } = 49$$

Solve for p

$$p=11$$

$$p=-3$$

Steps Using Factoring

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

$$p^{2}-8p+16=49$$

Subtract 49 from both sides.

$$p^{2}-8p+16-49=0$$

Subtract 49 from 16 to get -33.

$$p^{2}-8p-33=0$$

To solve the equation, factor p^{2}-8p-33 using formula p^{2}+\left(a+b\right)p+ab=\left(p+a\right)\left(p+b\right). To find a and b, set up a system to be solved.

$$a+b=-8 ab=-33$$

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 -33.

$$1,-33 3,-11$$

Calculate the sum for each pair.

$$1-33=-32 3-11=-8$$

The solution is the pair that gives sum -8.

$$a=-11 b=3$$

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

$$\left(p-11\right)\left(p+3\right)$$

To find equation solutions, solve p-11=0 and p+3=0.

$$p=11 p=-3$$

Steps Using Factoring By Grouping

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

$$p^{2}-8p+16=49$$

Subtract 49 from both sides.

$$p^{2}-8p+16-49=0$$

Subtract 49 from 16 to get -33.

$$p^{2}-8p-33=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as p^{2}+ap+bp-33. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\left(-33\right)=-33$$

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 -33.

$$1,-33 3,-11$$

Calculate the sum for each pair.

$$1-33=-32 3-11=-8$$

The solution is the pair that gives sum -8.

$$a=-11 b=3$$

Rewrite p^{2}-8p-33 as \left(p^{2}-11p\right)+\left(3p-33\right).

$$\left(p^{2}-11p\right)+\left(3p-33\right)$$

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

$$p\left(p-11\right)+3\left(p-11\right)$$

Factor out common term p-11 by using distributive property.

$$\left(p-11\right)\left(p+3\right)$$

To find equation solutions, solve p-11=0 and p+3=0.

$$p=11 p=-3$$

Steps Using the Quadratic Formula

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

$$p^{2}-8p+16=49$$

Subtract 49 from both sides.

$$p^{2}-8p+16-49=0$$

Subtract 49 from 16 to get -33.

$$p^{2}-8p-33=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and -33 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-33\right)}}{2}$$

Square -8.

$$p=\frac{-\left(-8\right)±\sqrt{64-4\left(-33\right)}}{2}$$

Multiply -4 times -33.

$$p=\frac{-\left(-8\right)±\sqrt{64+132}}{2}$$

Add 64 to 132.

$$p=\frac{-\left(-8\right)±\sqrt{196}}{2}$$

Take the square root of 196.

$$p=\frac{-\left(-8\right)±14}{2}$$

The opposite of -8 is 8.

$$p=\frac{8±14}{2}$$

Now solve the equation p=\frac{8±14}{2} when ± is plus. Add 8 to 14.

$$p=\frac{22}{2}$$

Divide 22 by 2.

$$p=11$$

Now solve the equation p=\frac{8±14}{2} when ± is minus. Subtract 14 from 8.

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

Divide -6 by 2.

$$p=-3$$

The equation is now solved.

$$p=11 p=-3$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(p-4\right)^{2}}=\sqrt{49}$$

Simplify.

$$p-4=7 p-4=-7$$

Add 4 to both sides of the equation.

$$p=11 p=-3$$