Solve 133x(x+2)+9=8x+1

$$133 x ( x + 2 ) + 9 = 8 x + 1$$

Solve for x

$$x=\frac{\sqrt{15577}-129}{133}\approx -0.031519907$$

$$x=\frac{-\sqrt{15577}-129}{133}\approx -1.908329717$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 133x by x+2.

$$133x^{2}+266x+9=8x+1$$

Subtract 8x from both sides.

$$133x^{2}+266x+9-8x=1$$

Combine 266x and -8x to get 258x.

$$133x^{2}+258x+9=1$$

Subtract 1 from both sides.

$$133x^{2}+258x+9-1=0$$

Subtract 1 from 9 to get 8.

$$133x^{2}+258x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 133 for a, 258 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-258±\sqrt{258^{2}-4\times 133\times 8}}{2\times 133}$$

Square 258.

$$x=\frac{-258±\sqrt{66564-4\times 133\times 8}}{2\times 133}$$

Multiply -4 times 133.

$$x=\frac{-258±\sqrt{66564-532\times 8}}{2\times 133}$$

Multiply -532 times 8.

$$x=\frac{-258±\sqrt{66564-4256}}{2\times 133}$$

Add 66564 to -4256.

$$x=\frac{-258±\sqrt{62308}}{2\times 133}$$

Take the square root of 62308.

$$x=\frac{-258±2\sqrt{15577}}{2\times 133}$$

Multiply 2 times 133.

$$x=\frac{-258±2\sqrt{15577}}{266}$$

Now solve the equation x=\frac{-258±2\sqrt{15577}}{266} when ± is plus. Add -258 to 2\sqrt{15577}.

$$x=\frac{2\sqrt{15577}-258}{266}$$

Divide -258+2\sqrt{15577} by 266.

$$x=\frac{\sqrt{15577}-129}{133}$$

Now solve the equation x=\frac{-258±2\sqrt{15577}}{266} when ± is minus. Subtract 2\sqrt{15577} from -258.

$$x=\frac{-2\sqrt{15577}-258}{266}$$

Divide -258-2\sqrt{15577} by 266.

$$x=\frac{-\sqrt{15577}-129}{133}$$

The equation is now solved.

$$x=\frac{\sqrt{15577}-129}{133} x=\frac{-\sqrt{15577}-129}{133}$$

Steps for Completing the Square

Use the distributive property to multiply 133x by x+2.

$$133x^{2}+266x+9=8x+1$$

Subtract 8x from both sides.

$$133x^{2}+266x+9-8x=1$$

Combine 266x and -8x to get 258x.

$$133x^{2}+258x+9=1$$

Subtract 9 from both sides.

$$133x^{2}+258x=1-9$$

Subtract 9 from 1 to get -8.

$$133x^{2}+258x=-8$$

Divide both sides by 133.

$$\frac{133x^{2}+258x}{133}=-\frac{8}{133}$$

Dividing by 133 undoes the multiplication by 133.

$$x^{2}+\frac{258}{133}x=-\frac{8}{133}$$

Divide \frac{258}{133}, the coefficient of the x term, by 2 to get \frac{129}{133}. Then add the square of \frac{129}{133} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{258}{133}x+\left(\frac{129}{133}\right)^{2}=-\frac{8}{133}+\left(\frac{129}{133}\right)^{2}$$

Square \frac{129}{133} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{258}{133}x+\frac{16641}{17689}=-\frac{8}{133}+\frac{16641}{17689}$$

Add -\frac{8}{133} to \frac{16641}{17689} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{258}{133}x+\frac{16641}{17689}=\frac{15577}{17689}$$

Factor x^{2}+\frac{258}{133}x+\frac{16641}{17689}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{129}{133}\right)^{2}=\frac{15577}{17689}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{129}{133}\right)^{2}}=\sqrt{\frac{15577}{17689}}$$

Simplify.

$$x+\frac{129}{133}=\frac{\sqrt{15577}}{133} x+\frac{129}{133}=-\frac{\sqrt{15577}}{133}$$

Subtract \frac{129}{133} from both sides of the equation.

$$x=\frac{\sqrt{15577}-129}{133} x=\frac{-\sqrt{15577}-129}{133}$$