Solve -6=sqrt{4x-24}-x

$$- 6 = \sqrt { 4 x - 24 } - x$$

Solve for x

$$x=10$$

$$x=6$$

Solution Steps

Subtract -x from both sides of the equation.

$$-6+x=\sqrt{4x-24}$$

Square both sides of the equation.

$$\left(-6+x\right)^{2}=\left(\sqrt{4x-24}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-6+x\right)^{2}.

$$36-12x+x^{2}=\left(\sqrt{4x-24}\right)^{2}$$

Calculate \sqrt{4x-24} to the power of 2 and get 4x-24.

$$36-12x+x^{2}=4x-24$$

Subtract 4x from both sides.

$$36-12x+x^{2}-4x=-24$$

Combine -12x and -4x to get -16x.

$$36-16x+x^{2}=-24$$

Add 24 to both sides.

$$36-16x+x^{2}+24=0$$

Add 36 and 24 to get 60.

$$60-16x+x^{2}=0$$

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

$$x^{2}-16x+60=0$$

To solve the equation, factor x^{2}-16x+60 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.

$$a+b=-16 ab=60$$

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 60.

$$-1,-60 -2,-30 -3,-20 -4,-15 -5,-12 -6,-10$$

Calculate the sum for each pair.

$$-1-60=-61 -2-30=-32 -3-20=-23 -4-15=-19 -5-12=-17 -6-10=-16$$

The solution is the pair that gives sum -16.

$$a=-10 b=-6$$

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

$$\left(x-10\right)\left(x-6\right)$$

To find equation solutions, solve x-10=0 and x-6=0.

$$x=10 x=6$$

Substitute 10 for x in the equation -6=\sqrt{4x-24}-x.

$$-6=\sqrt{4\times 10-24}-10$$

Simplify. The value x=10 satisfies the equation.

$$-6=-6$$

Substitute 6 for x in the equation -6=\sqrt{4x-24}-x.

$$-6=\sqrt{4\times 6-24}-6$$

Simplify. The value x=6 satisfies the equation.

$$-6=-6$$

List all solutions of x-6=\sqrt{4x-24}.

$$x=10 x=6$$