Solve sqrt{4x-3}=4x-5

$$\sqrt{ 4x-3 } = 4x-5$$

Solve for x

$$x = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{4x-3}\right)^{2}=\left(4x-5\right)^{2}$$

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

$$4x-3=\left(4x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-5\right)^{2}.

$$4x-3=16x^{2}-40x+25$$

Subtract 16x^{2} from both sides.

$$4x-3-16x^{2}=-40x+25$$

Add 40x to both sides.

$$4x-3-16x^{2}+40x=25$$

Combine 4x and 40x to get 44x.

$$44x-3-16x^{2}=25$$

Subtract 25 from both sides.

$$44x-3-16x^{2}-25=0$$

Subtract 25 from -3 to get -28.

$$44x-28-16x^{2}=0$$

Divide both sides by 4.

$$11x-7-4x^{2}=0$$

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

$$-4x^{2}+11x-7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=11 ab=-4\left(-7\right)=28$$

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 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 11.

$$a=7 b=4$$

Rewrite -4x^{2}+11x-7 as \left(-4x^{2}+7x\right)+\left(4x-7\right).

$$\left(-4x^{2}+7x\right)+\left(4x-7\right)$$

Factor out -x in -4x^{2}+7x.

$$-x\left(4x-7\right)+4x-7$$

Factor out common term 4x-7 by using distributive property.

$$\left(4x-7\right)\left(-x+1\right)$$

To find equation solutions, solve 4x-7=0 and -x+1=0.

$$x=\frac{7}{4} x=1$$

Substitute \frac{7}{4} for x in the equation \sqrt{4x-3}=4x-5.

$$\sqrt{4\times \frac{7}{4}-3}=4\times \frac{7}{4}-5$$

Simplify. The value x=\frac{7}{4} satisfies the equation.

$$2=2$$

Substitute 1 for x in the equation \sqrt{4x-3}=4x-5.

$$\sqrt{4\times 1-3}=4\times 1-5$$

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

$$1=-1$$

Equation \sqrt{4x-3}=4x-5 has a unique solution.

$$x=\frac{7}{4}$$