Solve -12=-x+sqrt{2x+24}

$$- 12 = - x + \sqrt { 2 x + 24 }$$

Solve for x

$$x=20$$

Solution Steps

Subtract -x from both sides of the equation.

$$-12-\left(-x\right)=\sqrt{2x+24}$$

Multiply -1 and -1 to get 1.

$$-12+x=\sqrt{2x+24}$$

Square both sides of the equation.

$$\left(-12+x\right)^{2}=\left(\sqrt{2x+24}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-12+x\right)^{2}.

$$144-24x+x^{2}=\left(\sqrt{2x+24}\right)^{2}$$

Calculate \sqrt{2x+24} to the power of 2 and get 2x+24.

$$144-24x+x^{2}=2x+24$$

Subtract 2x from both sides.

$$144-24x+x^{2}-2x=24$$

Combine -24x and -2x to get -26x.

$$144-26x+x^{2}=24$$

Subtract 24 from both sides.

$$144-26x+x^{2}-24=0$$

Subtract 24 from 144 to get 120.

$$120-26x+x^{2}=0$$

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

$$x^{2}-26x+120=0$$

To solve the equation, factor x^{2}-26x+120 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-26 ab=120$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 120.

$$-1,-120 -2,-60 -3,-40 -4,-30 -5,-24 -6,-20 -8,-15 -10,-12$$

Calculate the sum for each pair.

$$-1-120=-121 -2-60=-62 -3-40=-43 -4-30=-34 -5-24=-29 -6-20=-26 -8-15=-23 -10-12=-22$$

The solution is the pair that gives sum -26.

$$a=-20 b=-6$$

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

$$\left(x-20\right)\left(x-6\right)$$

To find equation solutions, solve x-20=0 and x-6=0.

$$x=20 x=6$$

Substitute 20 for x in the equation -12=-x+\sqrt{2x+24}.

$$-12=-20+\sqrt{2\times 20+24}$$

Simplify. The value x=20 satisfies the equation.

$$-12=-12$$

Substitute 6 for x in the equation -12=-x+\sqrt{2x+24}.

$$-12=-6+\sqrt{2\times 6+24}$$

Simplify. The value x=6 does not satisfy the equation.

$$-12=0$$

Equation x-12=\sqrt{2x+24} has a unique solution.

$$x=20$$