Solve sqrt{15-2x}=x

$$\sqrt { 15 - 2 x } = x$$

Solve for x

$$x=3$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{15-2x}\right)^{2}=x^{2}$$

Calculate \sqrt{15-2x} to the power of 2 and get 15-2x.

$$15-2x=x^{2}$$

Subtract x^{2} from both sides.

$$15-2x-x^{2}=0$$

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

$$-x^{2}-2x+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-15=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -15.

$$1,-15 3,-5$$

Calculate the sum for each pair.

$$1-15=-14 3-5=-2$$

The solution is the pair that gives sum -2.

$$a=3 b=-5$$

Rewrite -x^{2}-2x+15 as \left(-x^{2}+3x\right)+\left(-5x+15\right).

$$\left(-x^{2}+3x\right)+\left(-5x+15\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(-x+3\right)+5\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(x+5\right)$$

To find equation solutions, solve -x+3=0 and x+5=0.

$$x=3 x=-5$$

Substitute 3 for x in the equation \sqrt{15-2x}=x.

$$\sqrt{15-2\times 3}=3$$

Simplify. The value x=3 satisfies the equation.

$$3=3$$

Substitute -5 for x in the equation \sqrt{15-2x}=x.

$$\sqrt{15-2\left(-5\right)}=-5$$

Simplify. The value x=-5 does not satisfy the equation because the left and the right hand side have opposite signs.

$$5=-5$$

Equation \sqrt{15-2x}=x has a unique solution.

$$x=3$$