Solve 2x+sqrt{x+11}=14

$$2 x + \sqrt { x + 11 } = 14$$

Solve for x

$$x=5$$

Solution Steps

Subtract 2x from both sides of the equation.

$$\sqrt{x+11}=14-2x$$

Square both sides of the equation.

$$\left(\sqrt{x+11}\right)^{2}=\left(14-2x\right)^{2}$$

Calculate \sqrt{x+11} to the power of 2 and get x+11.

$$x+11=\left(14-2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(14-2x\right)^{2}.

$$x+11=196-56x+4x^{2}$$

Subtract 196 from both sides.

$$x+11-196=-56x+4x^{2}$$

Subtract 196 from 11 to get -185.

$$x-185=-56x+4x^{2}$$

Add 56x to both sides.

$$x-185+56x=4x^{2}$$

Combine x and 56x to get 57x.

$$57x-185=4x^{2}$$

Subtract 4x^{2} from both sides.

$$57x-185-4x^{2}=0$$

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

$$-4x^{2}+57x-185=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-185. To find a and b, set up a system to be solved.

$$a+b=57 ab=-4\left(-185\right)=740$$

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 740.

$$1,740 2,370 4,185 5,148 10,74 20,37$$

Calculate the sum for each pair.

$$1+740=741 2+370=372 4+185=189 5+148=153 10+74=84 20+37=57$$

The solution is the pair that gives sum 57.

$$a=37 b=20$$

Rewrite -4x^{2}+57x-185 as \left(-4x^{2}+37x\right)+\left(20x-185\right).

$$\left(-4x^{2}+37x\right)+\left(20x-185\right)$$

Factor out -x in the first and 5 in the second group.

$$-x\left(4x-37\right)+5\left(4x-37\right)$$

Factor out common term 4x-37 by using distributive property.

$$\left(4x-37\right)\left(-x+5\right)$$

To find equation solutions, solve 4x-37=0 and -x+5=0.

$$x=\frac{37}{4} x=5$$

Substitute \frac{37}{4} for x in the equation 2x+\sqrt{x+11}=14.

$$2\times \frac{37}{4}+\sqrt{\frac{37}{4}+11}=14$$

Simplify. The value x=\frac{37}{4} does not satisfy the equation.

$$23=14$$

Substitute 5 for x in the equation 2x+\sqrt{x+11}=14.

$$2\times 5+\sqrt{5+11}=14$$

Simplify. The value x=5 satisfies the equation.

$$14=14$$

Equation \sqrt{x+11}=14-2x has a unique solution.

$$x=5$$