Solve 2x-11sqrt{x+1}+7=0

$$2x-11 \sqrt{ x+1 } +7=0$$

Solve for x

$$x=24$$

$$x=-\frac{3}{4}=-0.75$$

Solution Steps

Subtract 7 from both sides. Anything subtracted from zero gives its negation.

$$2x-11\sqrt{x+1}=-7$$

Subtract 2x from both sides of the equation.

$$-11\sqrt{x+1}=-7-2x$$

Square both sides of the equation.

$$\left(-11\sqrt{x+1}\right)^{2}=\left(-7-2x\right)^{2}$$

Expand \left(-11\sqrt{x+1}\right)^{2}.

$$\left(-11\right)^{2}\left(\sqrt{x+1}\right)^{2}=\left(-7-2x\right)^{2}$$

Calculate -11 to the power of 2 and get 121.

$$121\left(\sqrt{x+1}\right)^{2}=\left(-7-2x\right)^{2}$$

Calculate \sqrt{x+1} to the power of 2 and get x+1.

$$121\left(x+1\right)=\left(-7-2x\right)^{2}$$

Use the distributive property to multiply 121 by x+1.

$$121x+121=\left(-7-2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-7-2x\right)^{2}.

$$121x+121=49+28x+4x^{2}$$

Subtract 28x from both sides.

$$121x+121-28x=49+4x^{2}$$

Combine 121x and -28x to get 93x.

$$93x+121=49+4x^{2}$$

Subtract 4x^{2} from both sides.

$$93x+121-4x^{2}=49$$

Subtract 49 from both sides.

$$93x+121-4x^{2}-49=0$$

Subtract 49 from 121 to get 72.

$$93x+72-4x^{2}=0$$

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

$$-4x^{2}+93x+72=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx+72. To find a and b, set up a system to be solved.

$$a+b=93 ab=-4\times 72=-288$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -288.

$$-1,288 -2,144 -3,96 -4,72 -6,48 -8,36 -9,32 -12,24 -16,18$$

Calculate the sum for each pair.

$$-1+288=287 -2+144=142 -3+96=93 -4+72=68 -6+48=42 -8+36=28 -9+32=23 -12+24=12 -16+18=2$$

The solution is the pair that gives sum 93.

$$a=96 b=-3$$

Rewrite -4x^{2}+93x+72 as \left(-4x^{2}+96x\right)+\left(-3x+72\right).

$$\left(-4x^{2}+96x\right)+\left(-3x+72\right)$$

Factor out 4x in the first and 3 in the second group.

$$4x\left(-x+24\right)+3\left(-x+24\right)$$

Factor out common term -x+24 by using distributive property.

$$\left(-x+24\right)\left(4x+3\right)$$

To find equation solutions, solve -x+24=0 and 4x+3=0.

$$x=24 x=-\frac{3}{4}$$

Substitute 24 for x in the equation 2x-11\sqrt{x+1}+7=0.

$$2\times 24-11\sqrt{24+1}+7=0$$

Simplify. The value x=24 satisfies the equation.

$$0=0$$

Substitute -\frac{3}{4} for x in the equation 2x-11\sqrt{x+1}+7=0.

$$2\left(-\frac{3}{4}\right)-11\sqrt{-\frac{3}{4}+1}+7=0$$

Simplify. The value x=-\frac{3}{4} satisfies the equation.

$$0=0$$

List all solutions of -11\sqrt{x+1}=-2x-7.

$$x=24 x=-\frac{3}{4}$$