Solve 2sqrt{x+11}=4x+2

$$2 \sqrt { x + 11 } = 4 x + 2$$

Solve for x

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Solution Steps

Square both sides of the equation.

$$\left(2\sqrt{x+11}\right)^{2}=\left(4x+2\right)^{2}$$

Expand \left(2\sqrt{x+11}\right)^{2}.

$$2^{2}\left(\sqrt{x+11}\right)^{2}=\left(4x+2\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x+11}\right)^{2}=\left(4x+2\right)^{2}$$

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

$$4\left(x+11\right)=\left(4x+2\right)^{2}$$

Use the distributive property to multiply 4 by x+11.

$$4x+44=\left(4x+2\right)^{2}$$

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

$$4x+44=16x^{2}+16x+4$$

Subtract 16x^{2} from both sides.

$$4x+44-16x^{2}=16x+4$$

Subtract 16x from both sides.

$$4x+44-16x^{2}-16x=4$$

Combine 4x and -16x to get -12x.

$$-12x+44-16x^{2}=4$$

Subtract 4 from both sides.

$$-12x+44-16x^{2}-4=0$$

Subtract 4 from 44 to get 40.

$$-12x+40-16x^{2}=0$$

Divide both sides by 4.

$$-3x+10-4x^{2}=0$$

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

$$-4x^{2}-3x+10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-4\times 10=-40$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -40.

$$1,-40 2,-20 4,-10 5,-8$$

Calculate the sum for each pair.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

The solution is the pair that gives sum -3.

$$a=5 b=-8$$

Rewrite -4x^{2}-3x+10 as \left(-4x^{2}+5x\right)+\left(-8x+10\right).

$$\left(-4x^{2}+5x\right)+\left(-8x+10\right)$$

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

$$-x\left(4x-5\right)-2\left(4x-5\right)$$

Factor out common term 4x-5 by using distributive property.

$$\left(4x-5\right)\left(-x-2\right)$$

To find equation solutions, solve 4x-5=0 and -x-2=0.

$$x=\frac{5}{4} x=-2$$

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

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

Simplify. The value x=\frac{5}{4} satisfies the equation.

$$7=7$$

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

$$2\sqrt{-2+11}=4\left(-2\right)+2$$

Simplify. The value x=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$6=-6$$

Equation 2\sqrt{x+11}=4x+2 has a unique solution.

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