Solve x-sqrt{6-x}=4

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

Solve for x

$$x=5$$

Solution Steps

Subtract x from both sides of the equation.

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

Square both sides of the equation.

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

Expand \left(-\sqrt{6-x}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{6-x}\right)^{2}=\left(4-x\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{6-x}\right)^{2}=\left(4-x\right)^{2}$$

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

$$1\left(6-x\right)=\left(4-x\right)^{2}$$

Use the distributive property to multiply 1 by 6-x.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-x\right)^{2}.

$$6-x=16-8x+x^{2}$$

Subtract 16 from both sides.

$$6-x-16=-8x+x^{2}$$

Subtract 16 from 6 to get -10.

$$-10-x=-8x+x^{2}$$

Add 8x to both sides.

$$-10-x+8x=x^{2}$$

Combine -x and 8x to get 7x.

$$-10+7x=x^{2}$$

Subtract x^{2} from both sides.

$$-10+7x-x^{2}=0$$

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

$$-x^{2}+7x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=7 ab=-\left(-10\right)=10$$

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 10.

$$1,10 2,5$$

Calculate the sum for each pair.

$$1+10=11 2+5=7$$

The solution is the pair that gives sum 7.

$$a=5 b=2$$

Rewrite -x^{2}+7x-10 as \left(-x^{2}+5x\right)+\left(2x-10\right).

$$\left(-x^{2}+5x\right)+\left(2x-10\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-5\right)+2\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(-x+2\right)$$

To find equation solutions, solve x-5=0 and -x+2=0.

$$x=5 x=2$$

Substitute 5 for x in the equation x-\sqrt{6-x}=4.

$$5-\sqrt{6-5}=4$$

Simplify. The value x=5 satisfies the equation.

$$4=4$$

Substitute 2 for x in the equation x-\sqrt{6-x}=4.

$$2-\sqrt{6-2}=4$$

Simplify. The value x=2 does not satisfy the equation.

$$0=4$$

Equation -\sqrt{6-x}=4-x has a unique solution.

$$x=5$$