Solve sqrt{-48+14x}=x

$$\sqrt{ -48+14x } =x$$

Solve for x

$$x=8$$

$$x=6$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{-48+14x}\right)^{2}=x^{2}$$

Calculate \sqrt{-48+14x} to the power of 2 and get -48+14x.

$$-48+14x=x^{2}$$

Subtract x^{2} from both sides.

$$-48+14x-x^{2}=0$$

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

$$-x^{2}+14x-48=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-48. To find a and b, set up a system to be solved.

$$a+b=14 ab=-\left(-48\right)=48$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 48.

$$1,48 2,24 3,16 4,12 6,8$$

Calculate the sum for each pair.

$$1+48=49 2+24=26 3+16=19 4+12=16 6+8=14$$

The solution is the pair that gives sum 14.

$$a=8 b=6$$

Rewrite -x^{2}+14x-48 as \left(-x^{2}+8x\right)+\left(6x-48\right).

$$\left(-x^{2}+8x\right)+\left(6x-48\right)$$

Factor out -x in the first and 6 in the second group.

$$-x\left(x-8\right)+6\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(-x+6\right)$$

To find equation solutions, solve x-8=0 and -x+6=0.

$$x=8 x=6$$

Substitute 8 for x in the equation \sqrt{-48+14x}=x.

$$\sqrt{-48+14\times 8}=8$$

Simplify. The value x=8 satisfies the equation.

$$8=8$$

Substitute 6 for x in the equation \sqrt{-48+14x}=x.

$$\sqrt{-48+14\times 6}=6$$

Simplify. The value x=6 satisfies the equation.

$$6=6$$

List all solutions of \sqrt{14x-48}=x.

$$x=8 x=6$$