Solve 3sqrt{x-1}+11=2x

$$3 \sqrt { x - 1 } + 11 = 2 x$$

Solve for x

$$x=10$$

Solution Steps

Subtract 11 from both sides of the equation.

$$3\sqrt{x-1}=2x-11$$

Square both sides of the equation.

$$\left(3\sqrt{x-1}\right)^{2}=\left(2x-11\right)^{2}$$

Expand \left(3\sqrt{x-1}\right)^{2}.

$$3^{2}\left(\sqrt{x-1}\right)^{2}=\left(2x-11\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{x-1}\right)^{2}=\left(2x-11\right)^{2}$$

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

$$9\left(x-1\right)=\left(2x-11\right)^{2}$$

Use the distributive property to multiply 9 by x-1.

$$9x-9=\left(2x-11\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-11\right)^{2}.

$$9x-9=4x^{2}-44x+121$$

Subtract 4x^{2} from both sides.

$$9x-9-4x^{2}=-44x+121$$

Add 44x to both sides.

$$9x-9-4x^{2}+44x=121$$

Combine 9x and 44x to get 53x.

$$53x-9-4x^{2}=121$$

Subtract 121 from both sides.

$$53x-9-4x^{2}-121=0$$

Subtract 121 from -9 to get -130.

$$53x-130-4x^{2}=0$$

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

$$-4x^{2}+53x-130=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-130. To find a and b, set up a system to be solved.

$$a+b=53 ab=-4\left(-130\right)=520$$

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 520.

$$1,520 2,260 4,130 5,104 8,65 10,52 13,40 20,26$$

Calculate the sum for each pair.

$$1+520=521 2+260=262 4+130=134 5+104=109 8+65=73 10+52=62 13+40=53 20+26=46$$

The solution is the pair that gives sum 53.

$$a=40 b=13$$

Rewrite -4x^{2}+53x-130 as \left(-4x^{2}+40x\right)+\left(13x-130\right).

$$\left(-4x^{2}+40x\right)+\left(13x-130\right)$$

Factor out 4x in the first and -13 in the second group.

$$4x\left(-x+10\right)-13\left(-x+10\right)$$

Factor out common term -x+10 by using distributive property.

$$\left(-x+10\right)\left(4x-13\right)$$

To find equation solutions, solve -x+10=0 and 4x-13=0.

$$x=10 x=\frac{13}{4}$$

Substitute 10 for x in the equation 3\sqrt{x-1}+11=2x.

$$3\sqrt{10-1}+11=2\times 10$$

Simplify. The value x=10 satisfies the equation.

$$20=20$$

Substitute \frac{13}{4} for x in the equation 3\sqrt{x-1}+11=2x.

$$3\sqrt{\frac{13}{4}-1}+11=2\times \frac{13}{4}$$

Simplify. The value x=\frac{13}{4} does not satisfy the equation.

$$\frac{31}{2}=\frac{13}{2}$$

Equation 3\sqrt{x-1}=2x-11 has a unique solution.

$$x=10$$