Solve sqrt{6+5x}=x

$$\sqrt { 6 + 5 x } = x$$

Solve for x

$$x=6$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{6+5x}\right)^{2}=x^{2}$$

Calculate \sqrt{6+5x} to the power of 2 and get 6+5x.

$$6+5x=x^{2}$$

Subtract x^{2} from both sides.

$$6+5x-x^{2}=0$$

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

$$-x^{2}+5x+6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=5 ab=-6=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 5.

$$a=6 b=-1$$

Rewrite -x^{2}+5x+6 as \left(-x^{2}+6x\right)+\left(-x+6\right).

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

Factor out -x in the first and -1 in the second group.

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

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(-x-1\right)$$

To find equation solutions, solve x-6=0 and -x-1=0.

$$x=6 x=-1$$

Substitute 6 for x in the equation \sqrt{6+5x}=x.

$$\sqrt{6+5\times 6}=6$$

Simplify. The value x=6 satisfies the equation.

$$6=6$$

Substitute -1 for x in the equation \sqrt{6+5x}=x.

$$\sqrt{6+5\left(-1\right)}=-1$$

Simplify. The value x=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Equation \sqrt{5x+6}=x has a unique solution.

$$x=6$$