Solve sqrt{80x+89}-3x=10

$$\sqrt { 80 x + 89 } - 3 x = 10$$

Solve for x

$$x = \frac{11}{9} = 1\frac{2}{9} \approx 1.222222222$$

$$x=1$$

Solution Steps

Subtract -3x from both sides of the equation.

$$\sqrt{80x+89}=10+3x$$

Square both sides of the equation.

$$\left(\sqrt{80x+89}\right)^{2}=\left(10+3x\right)^{2}$$

Calculate \sqrt{80x+89} to the power of 2 and get 80x+89.

$$80x+89=\left(10+3x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10+3x\right)^{2}.

$$80x+89=100+60x+9x^{2}$$

Subtract 100 from both sides.

$$80x+89-100=60x+9x^{2}$$

Subtract 100 from 89 to get -11.

$$80x-11=60x+9x^{2}$$

Subtract 60x from both sides.

$$80x-11-60x=9x^{2}$$

Combine 80x and -60x to get 20x.

$$20x-11=9x^{2}$$

Subtract 9x^{2} from both sides.

$$20x-11-9x^{2}=0$$

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

$$-9x^{2}+20x-11=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-11. To find a and b, set up a system to be solved.

$$a+b=20 ab=-9\left(-11\right)=99$$

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 99.

$$1,99 3,33 9,11$$

Calculate the sum for each pair.

$$1+99=100 3+33=36 9+11=20$$

The solution is the pair that gives sum 20.

$$a=11 b=9$$

Rewrite -9x^{2}+20x-11 as \left(-9x^{2}+11x\right)+\left(9x-11\right).

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

Factor out -x in -9x^{2}+11x.

$$-x\left(9x-11\right)+9x-11$$

Factor out common term 9x-11 by using distributive property.

$$\left(9x-11\right)\left(-x+1\right)$$

To find equation solutions, solve 9x-11=0 and -x+1=0.

$$x=\frac{11}{9} x=1$$

Substitute \frac{11}{9} for x in the equation \sqrt{80x+89}-3x=10.

$$\sqrt{80\times \frac{11}{9}+89}-3\times \frac{11}{9}=10$$

Simplify. The value x=\frac{11}{9} satisfies the equation.

$$10=10$$

Substitute 1 for x in the equation \sqrt{80x+89}-3x=10.

$$\sqrt{80\times 1+89}-3=10$$

Simplify. The value x=1 satisfies the equation.

$$10=10$$

List all solutions of \sqrt{80x+89}=3x+10.

$$x=\frac{11}{9} x=1$$