Solve 10x(x-1)=9x-6

$$10 x ( x - 1 ) = 9 x - 6$$

Solve for x

$$x=\frac{2}{5}=0.4$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10x by x-1.

$$10x^{2}-10x=9x-6$$

Subtract 9x from both sides.

$$10x^{2}-10x-9x=-6$$

Combine -10x and -9x to get -19x.

$$10x^{2}-19x=-6$$

Add 6 to both sides.

$$10x^{2}-19x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -19 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 10\times 6}}{2\times 10}$$

Square -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 10\times 6}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-19\right)±\sqrt{361-40\times 6}}{2\times 10}$$

Multiply -40 times 6.

$$x=\frac{-\left(-19\right)±\sqrt{361-240}}{2\times 10}$$

Add 361 to -240.

$$x=\frac{-\left(-19\right)±\sqrt{121}}{2\times 10}$$

Take the square root of 121.

$$x=\frac{-\left(-19\right)±11}{2\times 10}$$

The opposite of -19 is 19.

$$x=\frac{19±11}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{19±11}{20}$$

Now solve the equation x=\frac{19±11}{20} when ± is plus. Add 19 to 11.

$$x=\frac{30}{20}$$

Reduce the fraction \frac{30}{20} to lowest terms by extracting and canceling out 10.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{19±11}{20} when ± is minus. Subtract 11 from 19.

$$x=\frac{8}{20}$$

Reduce the fraction \frac{8}{20} to lowest terms by extracting and canceling out 4.

$$x=\frac{2}{5}$$

The equation is now solved.

$$x=\frac{3}{2} x=\frac{2}{5}$$

Steps for Completing the Square

Use the distributive property to multiply 10x by x-1.

$$10x^{2}-10x=9x-6$$

Subtract 9x from both sides.

$$10x^{2}-10x-9x=-6$$

Combine -10x and -9x to get -19x.

$$10x^{2}-19x=-6$$

Divide both sides by 10.

$$\frac{10x^{2}-19x}{10}=-\frac{6}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}-\frac{19}{10}x=-\frac{6}{10}$$

Reduce the fraction \frac{-6}{10} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{19}{10}x=-\frac{3}{5}$$

Divide -\frac{19}{10}, the coefficient of the x term, by 2 to get -\frac{19}{20}. Then add the square of -\frac{19}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{10}x+\left(-\frac{19}{20}\right)^{2}=-\frac{3}{5}+\left(-\frac{19}{20}\right)^{2}$$

Square -\frac{19}{20} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{10}x+\frac{361}{400}=-\frac{3}{5}+\frac{361}{400}$$

Add -\frac{3}{5} to \frac{361}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{19}{10}x+\frac{361}{400}=\frac{121}{400}$$

Factor x^{2}-\frac{19}{10}x+\frac{361}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{19}{20}\right)^{2}=\frac{121}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{20}\right)^{2}}=\sqrt{\frac{121}{400}}$$

Simplify.

$$x-\frac{19}{20}=\frac{11}{20} x-\frac{19}{20}=-\frac{11}{20}$$

Add \frac{19}{20} to both sides of the equation.

$$x=\frac{3}{2} x=\frac{2}{5}$$