Solve -6p+2sqrt{p^2+8}=0

$$- 6 p + 2 \sqrt { p ^ { 2 } + 8 } = 0$$

Solve for p

$$p=1$$

Solution Steps

Subtract -6p from both sides of the equation.

$$2\sqrt{p^{2}+8}=6p$$

Square both sides of the equation.

$$\left(2\sqrt{p^{2}+8}\right)^{2}=\left(6p\right)^{2}$$

Expand \left(2\sqrt{p^{2}+8}\right)^{2}.

$$2^{2}\left(\sqrt{p^{2}+8}\right)^{2}=\left(6p\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{p^{2}+8}\right)^{2}=\left(6p\right)^{2}$$

Calculate \sqrt{p^{2}+8} to the power of 2 and get p^{2}+8.

$$4\left(p^{2}+8\right)=\left(6p\right)^{2}$$

Use the distributive property to multiply 4 by p^{2}+8.

$$4p^{2}+32=\left(6p\right)^{2}$$

Expand \left(6p\right)^{2}.

$$4p^{2}+32=6^{2}p^{2}$$

Calculate 6 to the power of 2 and get 36.

$$4p^{2}+32=36p^{2}$$

Subtract 36p^{2} from both sides.

$$4p^{2}+32-36p^{2}=0$$

Combine 4p^{2} and -36p^{2} to get -32p^{2}.

$$-32p^{2}+32=0$$

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

$$-32p^{2}=-32$$

Divide both sides by -32.

$$p^{2}=\frac{-32}{-32}$$

Divide -32 by -32 to get 1.

$$p^{2}=1$$

Take the square root of both sides of the equation.

$$p=1 p=-1$$

Substitute 1 for p in the equation -6p+2\sqrt{p^{2}+8}=0.

$$-6+2\sqrt{1^{2}+8}=0$$

Simplify. The value p=1 satisfies the equation.

$$0=0$$

Substitute -1 for p in the equation -6p+2\sqrt{p^{2}+8}=0.

$$-6\left(-1\right)+2\sqrt{\left(-1\right)^{2}+8}=0$$

Simplify. The value p=-1 does not satisfy the equation.

$$12=0$$

Equation 2\sqrt{p^{2}+8}=6p has a unique solution.

$$p=1$$