Solve sqrt{4x-3}=x

$$\sqrt{ 4x-3 } =x$$

Solve for x

$$x=3$$

$$x=1$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{4x-3}\right)^{2}=x^{2}$$

Calculate \sqrt{4x-3} to the power of 2 and get 4x-3.

$$4x-3=x^{2}$$

Subtract x^{2} from both sides.

$$4x-3-x^{2}=0$$

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

$$-x^{2}+4x-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=4 ab=-\left(-3\right)=3$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=3 b=1$$

Rewrite -x^{2}+4x-3 as \left(-x^{2}+3x\right)+\left(x-3\right).

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

Factor out -x in -x^{2}+3x.

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

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

$$\left(x-3\right)\left(-x+1\right)$$

To find equation solutions, solve x-3=0 and -x+1=0.

$$x=3 x=1$$

Substitute 3 for x in the equation \sqrt{4x-3}=x.

$$\sqrt{4\times 3-3}=3$$

Simplify. The value x=3 satisfies the equation.

$$3=3$$

Substitute 1 for x in the equation \sqrt{4x-3}=x.

$$\sqrt{4\times 1-3}=1$$

Simplify. The value x=1 satisfies the equation.

$$1=1$$

List all solutions of \sqrt{4x-3}=x.

$$x=3 x=1$$