Solve x(2x-1)-3(x-5)=(3x-1)(2+x)-5.11

$$x ( 2 x - 1 ) - 3 ( x - 5 ) = ( 3 x - 1 ) ( 2 + x ) - 5.11$$

Solve for x

$$x=\frac{\sqrt{1059}}{5}-4.5\approx 2.008456038$$

$$x=-\frac{\sqrt{1059}}{5}-4.5\approx -11.008456038$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 2x-1.

$$2x^{2}-x-3\left(x-5\right)=\left(3x-1\right)\left(2+x\right)-5.11$$

Use the distributive property to multiply -3 by x-5.

$$2x^{2}-x-3x+15=\left(3x-1\right)\left(2+x\right)-5.11$$

Combine -x and -3x to get -4x.

$$2x^{2}-4x+15=\left(3x-1\right)\left(2+x\right)-5.11$$

Use the distributive property to multiply 3x-1 by 2+x and combine like terms.

$$2x^{2}-4x+15=5x+3x^{2}-2-5.11$$

Subtract 5.11 from -2 to get -7.11.

$$2x^{2}-4x+15=5x+3x^{2}-7.11$$

Subtract 5x from both sides.

$$2x^{2}-4x+15-5x=3x^{2}-7.11$$

Combine -4x and -5x to get -9x.

$$2x^{2}-9x+15=3x^{2}-7.11$$

Subtract 3x^{2} from both sides.

$$2x^{2}-9x+15-3x^{2}=-7.11$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}-9x+15=-7.11$$

Add 7.11 to both sides.

$$-x^{2}-9x+15+7.11=0$$

Add 15 and 7.11 to get 22.11.

$$-x^{2}-9x+22.11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -9 for b, and 22.11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-1\right)\times 22.11}}{2\left(-1\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-1\right)\times 22.11}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-9\right)±\sqrt{81+4\times 22.11}}{2\left(-1\right)}$$

Multiply 4 times 22.11.

$$x=\frac{-\left(-9\right)±\sqrt{81+88.44}}{2\left(-1\right)}$$

Add 81 to 88.44.

$$x=\frac{-\left(-9\right)±\sqrt{169.44}}{2\left(-1\right)}$$

Take the square root of 169.44.

$$x=\frac{-\left(-9\right)±\frac{2\sqrt{1059}}{5}}{2\left(-1\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±\frac{2\sqrt{1059}}{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{9±\frac{2\sqrt{1059}}{5}}{-2}$$

Now solve the equation x=\frac{9±\frac{2\sqrt{1059}}{5}}{-2} when ± is plus. Add 9 to \frac{2\sqrt{1059}}{5}.

$$x=\frac{\frac{2\sqrt{1059}}{5}+9}{-2}$$

Divide 9+\frac{2\sqrt{1059}}{5} by -2.

$$x=-\frac{\sqrt{1059}}{5}-\frac{9}{2}$$

Now solve the equation x=\frac{9±\frac{2\sqrt{1059}}{5}}{-2} when ± is minus. Subtract \frac{2\sqrt{1059}}{5} from 9.

$$x=\frac{-\frac{2\sqrt{1059}}{5}+9}{-2}$$

Divide 9-\frac{2\sqrt{1059}}{5} by -2.

$$x=\frac{\sqrt{1059}}{5}-\frac{9}{2}$$

The equation is now solved.

$$x=-\frac{\sqrt{1059}}{5}-\frac{9}{2} x=\frac{\sqrt{1059}}{5}-\frac{9}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x by 2x-1.

$$2x^{2}-x-3\left(x-5\right)=\left(3x-1\right)\left(2+x\right)-5.11$$

Use the distributive property to multiply -3 by x-5.

$$2x^{2}-x-3x+15=\left(3x-1\right)\left(2+x\right)-5.11$$

Combine -x and -3x to get -4x.

$$2x^{2}-4x+15=\left(3x-1\right)\left(2+x\right)-5.11$$

Use the distributive property to multiply 3x-1 by 2+x and combine like terms.

$$2x^{2}-4x+15=5x+3x^{2}-2-5.11$$

Subtract 5.11 from -2 to get -7.11.

$$2x^{2}-4x+15=5x+3x^{2}-7.11$$

Subtract 5x from both sides.

$$2x^{2}-4x+15-5x=3x^{2}-7.11$$

Combine -4x and -5x to get -9x.

$$2x^{2}-9x+15=3x^{2}-7.11$$

Subtract 3x^{2} from both sides.

$$2x^{2}-9x+15-3x^{2}=-7.11$$

Combine 2x^{2} and -3x^{2} to get -x^{2}.

$$-x^{2}-9x+15=-7.11$$

Subtract 15 from both sides.

$$-x^{2}-9x=-7.11-15$$

Subtract 15 from -7.11 to get -22.11.

$$-x^{2}-9x=-22.11$$

Divide both sides by -1.

$$\frac{-x^{2}-9x}{-1}=-\frac{22.11}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{9}{-1}\right)x=-\frac{22.11}{-1}$$

Divide -9 by -1.

$$x^{2}+9x=-\frac{22.11}{-1}$$

Divide -22.11 by -1.

$$x^{2}+9x=22.11$$

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=22.11+\left(\frac{9}{2}\right)^{2}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+9x+\frac{81}{4}=22.11+\frac{81}{4}$$

Add 22.11 to \frac{81}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+9x+\frac{81}{4}=\frac{1059}{25}$$

Factor x^{2}+9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{1059}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{1059}{25}}$$

Simplify.

$$x+\frac{9}{2}=\frac{\sqrt{1059}}{5} x+\frac{9}{2}=-\frac{\sqrt{1059}}{5}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=\frac{\sqrt{1059}}{5}-\frac{9}{2} x=-\frac{\sqrt{1059}}{5}-\frac{9}{2}$$