Solve (p-7)(p+3)=24

$$( p - 7 ) ( p + 3 ) = 24$$

Solve for p

$$p=-5$$

$$p=9$$

Steps Using the Quadratic Formula

Use the distributive property to multiply p-7 by p+3 and combine like terms.

$$p^{2}-4p-21=24$$

Subtract 24 from both sides.

$$p^{2}-4p-21-24=0$$

Subtract 24 from -21 to get -45.

$$p^{2}-4p-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-45\right)}}{2}$$

Square -4.

$$p=\frac{-\left(-4\right)±\sqrt{16-4\left(-45\right)}}{2}$$

Multiply -4 times -45.

$$p=\frac{-\left(-4\right)±\sqrt{16+180}}{2}$$

Add 16 to 180.

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

Take the square root of 196.

$$p=\frac{-\left(-4\right)±14}{2}$$

The opposite of -4 is 4.

$$p=\frac{4±14}{2}$$

Now solve the equation p=\frac{4±14}{2} when ± is plus. Add 4 to 14.

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

Divide 18 by 2.

$$p=9$$

Now solve the equation p=\frac{4±14}{2} when ± is minus. Subtract 14 from 4.

$$p=-\frac{10}{2}$$

Divide -10 by 2.

$$p=-5$$

The equation is now solved.

$$p=9 p=-5$$

Steps for Completing the Square

Use the distributive property to multiply p-7 by p+3 and combine like terms.

$$p^{2}-4p-21=24$$

Add 21 to both sides.

$$p^{2}-4p=24+21$$

Add 24 and 21 to get 45.

$$p^{2}-4p=45$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-4p+\left(-2\right)^{2}=45+\left(-2\right)^{2}$$

Square -2.

$$p^{2}-4p+4=45+4$$

Add 45 to 4.

$$p^{2}-4p+4=49$$

Factor p^{2}-4p+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$p-2=7 p-2=-7$$

Add 2 to both sides of the equation.

$$p=9 p=-5$$