Solve sqrt{63-2x}-x=0

$$\sqrt { 63 - 2 x } - x = 0$$

Solve for x

$$x=7$$

Solution Steps

Subtract -x from both sides of the equation.

$$\sqrt{63-2x}=x$$

Square both sides of the equation.

$$\left(\sqrt{63-2x}\right)^{2}=x^{2}$$

Calculate \sqrt{63-2x} to the power of 2 and get 63-2x.

$$63-2x=x^{2}$$

Subtract x^{2} from both sides.

$$63-2x-x^{2}=0$$

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

$$-x^{2}-2x+63=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+63. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-63=-63$$

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 -63.

$$1,-63 3,-21 7,-9$$

Calculate the sum for each pair.

$$1-63=-62 3-21=-18 7-9=-2$$

The solution is the pair that gives sum -2.

$$a=7 b=-9$$

Rewrite -x^{2}-2x+63 as \left(-x^{2}+7x\right)+\left(-9x+63\right).

$$\left(-x^{2}+7x\right)+\left(-9x+63\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(-x+7\right)+9\left(-x+7\right)$$

Factor out common term -x+7 by using distributive property.

$$\left(-x+7\right)\left(x+9\right)$$

To find equation solutions, solve -x+7=0 and x+9=0.

$$x=7 x=-9$$

Substitute 7 for x in the equation \sqrt{63-2x}-x=0.

$$\sqrt{63-2\times 7}-7=0$$

Simplify. The value x=7 satisfies the equation.

$$0=0$$

Substitute -9 for x in the equation \sqrt{63-2x}-x=0.

$$\sqrt{63-2\left(-9\right)}-\left(-9\right)=0$$

Simplify. The value x=-9 does not satisfy the equation.

$$18=0$$

Equation \sqrt{63-2x}=x has a unique solution.

$$x=7$$