Solve x-sqrt{3x-11}=7

$$x - \sqrt { 3 x - 11 } = 7$$

Solve for x

$$x=12$$

Solution Steps

Subtract x from both sides of the equation.

$$-\sqrt{3x-11}=7-x$$

Square both sides of the equation.

$$\left(-\sqrt{3x-11}\right)^{2}=\left(7-x\right)^{2}$$

Expand \left(-\sqrt{3x-11}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{3x-11}\right)^{2}=\left(7-x\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{3x-11}\right)^{2}=\left(7-x\right)^{2}$$

Calculate \sqrt{3x-11} to the power of 2 and get 3x-11.

$$1\left(3x-11\right)=\left(7-x\right)^{2}$$

Use the distributive property to multiply 1 by 3x-11.

$$3x-11=\left(7-x\right)^{2}$$

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

$$3x-11=49-14x+x^{2}$$

Subtract 49 from both sides.

$$3x-11-49=-14x+x^{2}$$

Subtract 49 from -11 to get -60.

$$3x-60=-14x+x^{2}$$

Add 14x to both sides.

$$3x-60+14x=x^{2}$$

Combine 3x and 14x to get 17x.

$$17x-60=x^{2}$$

Subtract x^{2} from both sides.

$$17x-60-x^{2}=0$$

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

$$-x^{2}+17x-60=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-60. To find a and b, set up a system to be solved.

$$a+b=17 ab=-\left(-60\right)=60$$

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 60.

$$1,60 2,30 3,20 4,15 5,12 6,10$$

Calculate the sum for each pair.

$$1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16$$

The solution is the pair that gives sum 17.

$$a=12 b=5$$

Rewrite -x^{2}+17x-60 as \left(-x^{2}+12x\right)+\left(5x-60\right).

$$\left(-x^{2}+12x\right)+\left(5x-60\right)$$

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

$$-x\left(x-12\right)+5\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(-x+5\right)$$

To find equation solutions, solve x-12=0 and -x+5=0.

$$x=12 x=5$$

Substitute 12 for x in the equation x-\sqrt{3x-11}=7.

$$12-\sqrt{3\times 12-11}=7$$

Simplify. The value x=12 satisfies the equation.

$$7=7$$

Substitute 5 for x in the equation x-\sqrt{3x-11}=7.

$$5-\sqrt{3\times 5-11}=7$$

Simplify. The value x=5 does not satisfy the equation.

$$3=7$$

Equation -\sqrt{3x-11}=7-x has a unique solution.

$$x=12$$