Solve 6x(x-1)=21-x

$$6 x ( x - 1 ) = 21 - x$$

Solve for x

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

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6x by x-1.

$$6x^{2}-6x=21-x$$

Subtract 21 from both sides.

$$6x^{2}-6x-21=-x$$

Add x to both sides.

$$6x^{2}-6x-21+x=0$$

Combine -6x and x to get -5x.

$$6x^{2}-5x-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -5 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6\left(-21\right)}}{2\times 6}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 6\left(-21\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-5\right)±\sqrt{25-24\left(-21\right)}}{2\times 6}$$

Multiply -24 times -21.

$$x=\frac{-\left(-5\right)±\sqrt{25+504}}{2\times 6}$$

Add 25 to 504.

$$x=\frac{-\left(-5\right)±\sqrt{529}}{2\times 6}$$

Take the square root of 529.

$$x=\frac{-\left(-5\right)±23}{2\times 6}$$

The opposite of -5 is 5.

$$x=\frac{5±23}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{5±23}{12}$$

Now solve the equation x=\frac{5±23}{12} when ± is plus. Add 5 to 23.

$$x=\frac{28}{12}$$

Reduce the fraction \frac{28}{12} to lowest terms by extracting and canceling out 4.

$$x=\frac{7}{3}$$

Now solve the equation x=\frac{5±23}{12} when ± is minus. Subtract 23 from 5.

$$x=-\frac{18}{12}$$

Reduce the fraction \frac{-18}{12} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

$$x=\frac{7}{3} x=-\frac{3}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 6x by x-1.

$$6x^{2}-6x=21-x$$

Add x to both sides.

$$6x^{2}-6x+x=21$$

Combine -6x and x to get -5x.

$$6x^{2}-5x=21$$

Divide both sides by 6.

$$\frac{6x^{2}-5x}{6}=\frac{21}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{5}{6}x=\frac{21}{6}$$

Reduce the fraction \frac{21}{6} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{5}{6}x=\frac{7}{2}$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=\frac{7}{2}+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{7}{2}+\frac{25}{144}$$

Add \frac{7}{2} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{529}{144}$$

Factor x^{2}-\frac{5}{6}x+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{12}\right)^{2}=\frac{529}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{\frac{529}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{23}{12} x-\frac{5}{12}=-\frac{23}{12}$$

Add \frac{5}{12} to both sides of the equation.

$$x=\frac{7}{3} x=-\frac{3}{2}$$