Solve 12-p^2=6p-4

$$12- { p }^{ 2 } = 6p-4$$

Solve for p

$$p=-8$$

$$p=2$$

Steps Using Factoring By Grouping

Subtract 6p from both sides.

$$12-p^{2}-6p=-4$$

Add 4 to both sides.

$$12-p^{2}-6p+4=0$$

Add 12 and 4 to get 16.

$$16-p^{2}-6p=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-p^{2}-6p+16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -p^{2}+ap+bp+16. To find a and b, set up a system to be solved.

$$a+b=-6 ab=-16=-16$$

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 -16.

$$1,-16 2,-8 4,-4$$

Calculate the sum for each pair.

$$1-16=-15 2-8=-6 4-4=0$$

The solution is the pair that gives sum -6.

$$a=2 b=-8$$

Rewrite -p^{2}-6p+16 as \left(-p^{2}+2p\right)+\left(-8p+16\right).

$$\left(-p^{2}+2p\right)+\left(-8p+16\right)$$

Factor out p in the first and 8 in the second group.

$$p\left(-p+2\right)+8\left(-p+2\right)$$

Factor out common term -p+2 by using distributive property.

$$\left(-p+2\right)\left(p+8\right)$$

To find equation solutions, solve -p+2=0 and p+8=0.

$$p=2 p=-8$$

Steps Using the Quadratic Formula

Subtract 6p from both sides.

$$12-p^{2}-6p=-4$$

Add 4 to both sides.

$$12-p^{2}-6p+4=0$$

Add 12 and 4 to get 16.

$$16-p^{2}-6p=0$$

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.

$$-p^{2}-6p+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -6 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Square -6.

$$p=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$p=\frac{-\left(-6\right)±\sqrt{36+4\times 16}}{2\left(-1\right)}$$

Multiply 4 times 16.

$$p=\frac{-\left(-6\right)±\sqrt{36+64}}{2\left(-1\right)}$$

Add 36 to 64.

$$p=\frac{-\left(-6\right)±\sqrt{100}}{2\left(-1\right)}$$

Take the square root of 100.

$$p=\frac{-\left(-6\right)±10}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$p=\frac{6±10}{2\left(-1\right)}$$

Multiply 2 times -1.

$$p=\frac{6±10}{-2}$$

Now solve the equation p=\frac{6±10}{-2} when ± is plus. Add 6 to 10.

$$p=\frac{16}{-2}$$

Divide 16 by -2.

$$p=-8$$

Now solve the equation p=\frac{6±10}{-2} when ± is minus. Subtract 10 from 6.

$$p=-\frac{4}{-2}$$

Divide -4 by -2.

$$p=2$$

The equation is now solved.

$$p=-8 p=2$$

Steps for Completing the Square

Subtract 6p from both sides.

$$12-p^{2}-6p=-4$$

Subtract 12 from both sides.

$$-p^{2}-6p=-4-12$$

Subtract 12 from -4 to get -16.

$$-p^{2}-6p=-16$$

Divide both sides by -1.

$$\frac{-p^{2}-6p}{-1}=-\frac{16}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$p^{2}+\left(-\frac{6}{-1}\right)p=-\frac{16}{-1}$$

Divide -6 by -1.

$$p^{2}+6p=-\frac{16}{-1}$$

Divide -16 by -1.

$$p^{2}+6p=16$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+6p+3^{2}=16+3^{2}$$

Square 3.

$$p^{2}+6p+9=16+9$$

Add 16 to 9.

$$p^{2}+6p+9=25$$

Factor p^{2}+6p+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+3\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+3\right)^{2}}=\sqrt{25}$$

Simplify.

$$p+3=5 p+3=-5$$

Subtract 3 from both sides of the equation.

$$p=2 p=-8$$