Solve sqrt{4x+5}-6=2x-11

$$\sqrt { 4 x + 5 } - 6 = 2 x - 11$$

Solve for x

$$x=5$$

Solution Steps

Subtract -6 from both sides of the equation.

$$\sqrt{4x+5}=2x-11+6$$

Add -11 and 6 to get -5.

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

Square both sides of the equation.

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

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

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

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

$$4x+5=4x^{2}-20x+25$$

Subtract 4x^{2} from both sides.

$$4x+5-4x^{2}=-20x+25$$

Add 20x to both sides.

$$4x+5-4x^{2}+20x=25$$

Combine 4x and 20x to get 24x.

$$24x+5-4x^{2}=25$$

Subtract 25 from both sides.

$$24x+5-4x^{2}-25=0$$

Subtract 25 from 5 to get -20.

$$24x-20-4x^{2}=0$$

Divide both sides by 4.

$$6x-5-x^{2}=0$$

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

$$-x^{2}+6x-5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=6 ab=-\left(-5\right)=5$$

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.

$$a=5 b=1$$

Rewrite -x^{2}+6x-5 as \left(-x^{2}+5x\right)+\left(x-5\right).

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

Factor out -x in -x^{2}+5x.

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

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

$$\left(x-5\right)\left(-x+1\right)$$

To find equation solutions, solve x-5=0 and -x+1=0.

$$x=5 x=1$$

Substitute 5 for x in the equation \sqrt{4x+5}-6=2x-11.

$$\sqrt{4\times 5+5}-6=2\times 5-11$$

Simplify. The value x=5 satisfies the equation.

$$-1=-1$$

Substitute 1 for x in the equation \sqrt{4x+5}-6=2x-11.

$$\sqrt{4\times 1+5}-6=2\times 1-11$$

Simplify. The value x=1 does not satisfy the equation.

$$-3=-9$$

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

$$x=5$$