Solve sqrt{-2x+5}=2x-3

$$\sqrt{ -2x+5 } =2x-3$$

Solve for x

$$x=2$$

Solution Steps

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-3\right)^{2}.

$$-2x+5=4x^{2}-12x+9$$

Subtract 4x^{2} from both sides.

$$-2x+5-4x^{2}=-12x+9$$

Add 12x to both sides.

$$-2x+5-4x^{2}+12x=9$$

Combine -2x and 12x to get 10x.

$$10x+5-4x^{2}=9$$

Subtract 9 from both sides.

$$10x+5-4x^{2}-9=0$$

Subtract 9 from 5 to get -4.

$$10x-4-4x^{2}=0$$

Divide both sides by 2.

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

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

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

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

$$a+b=5 ab=-2\left(-2\right)=4$$

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 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 5.

$$a=4 b=1$$

Rewrite -2x^{2}+5x-2 as \left(-2x^{2}+4x\right)+\left(x-2\right).

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

Factor out 2x in the first and -1 in the second group.

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

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(2x-1\right)$$

To find equation solutions, solve -x+2=0 and 2x-1=0.

$$x=2 x=\frac{1}{2}$$

Substitute 2 for x in the equation \sqrt{-2x+5}=2x-3.

$$\sqrt{-2\times 2+5}=2\times 2-3$$

Simplify. The value x=2 satisfies the equation.

$$1=1$$

Substitute \frac{1}{2} for x in the equation \sqrt{-2x+5}=2x-3.

$$\sqrt{-2\times \frac{1}{2}+5}=2\times \frac{1}{2}-3$$

Simplify. The value x=\frac{1}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$2=-2$$

Equation \sqrt{5-2x}=2x-3 has a unique solution.

$$x=2$$