Solve sqrt{-12x+73}-3x=1

$$\sqrt { - 12 x + 73 } - 3 x = 1$$

Solve for x

$$x=2$$

Solution Steps

Subtract -3x from both sides of the equation.

$$\sqrt{-12x+73}=1+3x$$

Square both sides of the equation.

$$\left(\sqrt{-12x+73}\right)^{2}=\left(1+3x\right)^{2}$$

Calculate \sqrt{-12x+73} to the power of 2 and get -12x+73.

$$-12x+73=\left(1+3x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+3x\right)^{2}.

$$-12x+73=1+6x+9x^{2}$$

Subtract 1 from both sides.

$$-12x+73-1=6x+9x^{2}$$

Subtract 1 from 73 to get 72.

$$-12x+72=6x+9x^{2}$$

Subtract 6x from both sides.

$$-12x+72-6x=9x^{2}$$

Combine -12x and -6x to get -18x.

$$-18x+72=9x^{2}$$

Subtract 9x^{2} from both sides.

$$-18x+72-9x^{2}=0$$

Divide both sides by 9.

$$-2x+8-x^{2}=0$$

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

$$-x^{2}-2x+8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-8=-8$$

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 -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -2.

$$a=2 b=-4$$

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

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

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

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

Factor out common term -x+2 by using distributive property.

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

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

$$x=2 x=-4$$

Substitute 2 for x in the equation \sqrt{-12x+73}-3x=1.

$$\sqrt{-12\times 2+73}-3\times 2=1$$

Simplify. The value x=2 satisfies the equation.

$$1=1$$

Substitute -4 for x in the equation \sqrt{-12x+73}-3x=1.

$$\sqrt{-12\left(-4\right)+73}-3\left(-4\right)=1$$

Simplify. The value x=-4 does not satisfy the equation.

$$23=1$$

Equation \sqrt{73-12x}=3x+1 has a unique solution.

$$x=2$$