Solve sqrt{5x-4}+x-2=0

$$\sqrt { 5 x - 4 } + x - 2 = 0$$

Solve for x

$$x=1$$

Solution Steps

Subtract x-2 from both sides of the equation.

$$\sqrt{5x-4}=-\left(x-2\right)$$

To find the opposite of x-2, find the opposite of each term.

$$\sqrt{5x-4}=-x-\left(-2\right)$$

The opposite of -2 is 2.

$$\sqrt{5x-4}=-x+2$$

Square both sides of the equation.

$$\left(\sqrt{5x-4}\right)^{2}=\left(-x+2\right)^{2}$$

Calculate \sqrt{5x-4} to the power of 2 and get 5x-4.

$$5x-4=\left(-x+2\right)^{2}$$

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

$$5x-4=x^{2}-4x+4$$

Subtract x^{2} from both sides.

$$5x-4-x^{2}=-4x+4$$

Add 4x to both sides.

$$5x-4-x^{2}+4x=4$$

Combine 5x and 4x to get 9x.

$$9x-4-x^{2}=4$$

Subtract 4 from both sides.

$$9x-4-x^{2}-4=0$$

Subtract 4 from -4 to get -8.

$$9x-8-x^{2}=0$$

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

$$-x^{2}+9x-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-8. To find a and b, set up a system to be solved.

$$a+b=9 ab=-\left(-8\right)=8$$

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 8.

$$1,8 2,4$$

Calculate the sum for each pair.

$$1+8=9 2+4=6$$

The solution is the pair that gives sum 9.

$$a=8 b=1$$

Rewrite -x^{2}+9x-8 as \left(-x^{2}+8x\right)+\left(x-8\right).

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

Factor out -x in -x^{2}+8x.

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

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

$$\left(x-8\right)\left(-x+1\right)$$

To find equation solutions, solve x-8=0 and -x+1=0.

$$x=8 x=1$$

Substitute 8 for x in the equation \sqrt{5x-4}+x-2=0.

$$\sqrt{5\times 8-4}+8-2=0$$

Simplify. The value x=8 does not satisfy the equation.

$$12=0$$

Substitute 1 for x in the equation \sqrt{5x-4}+x-2=0.

$$\sqrt{5\times 1-4}+1-2=0$$

Simplify. The value x=1 satisfies the equation.

$$0=0$$

Equation \sqrt{5x-4}=2-x has a unique solution.

$$x=1$$