Solve 66x(x-1)=21x

$$66 x ( x - 1 ) = 21 x$$

Solve for x

$$x = \frac{29}{22} = 1\frac{7}{22} \approx 1.318181818$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 66x by x-1.

$$66x^{2}-66x=21x$$

Subtract 21x from both sides.

$$66x^{2}-66x-21x=0$$

Combine -66x and -21x to get -87x.

$$66x^{2}-87x=0$$

Factor out x.

$$x\left(66x-87\right)=0$$

To find equation solutions, solve x=0 and 66x-87=0.

$$x=0 x=\frac{29}{22}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 66x by x-1.

$$66x^{2}-66x=21x$$

Subtract 21x from both sides.

$$66x^{2}-66x-21x=0$$

Combine -66x and -21x to get -87x.

$$66x^{2}-87x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 66 for a, -87 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-87\right)±\sqrt{\left(-87\right)^{2}}}{2\times 66}$$

Take the square root of \left(-87\right)^{2}.

$$x=\frac{-\left(-87\right)±87}{2\times 66}$$

The opposite of -87 is 87.

$$x=\frac{87±87}{2\times 66}$$

Multiply 2 times 66.

$$x=\frac{87±87}{132}$$

Now solve the equation x=\frac{87±87}{132} when ± is plus. Add 87 to 87.

$$x=\frac{174}{132}$$

Reduce the fraction \frac{174}{132} to lowest terms by extracting and canceling out 6.

$$x=\frac{29}{22}$$

Now solve the equation x=\frac{87±87}{132} when ± is minus. Subtract 87 from 87.

$$x=\frac{0}{132}$$

Divide 0 by 132.

$$x=0$$

The equation is now solved.

$$x=\frac{29}{22} x=0$$

Steps for Completing the Square

Use the distributive property to multiply 66x by x-1.

$$66x^{2}-66x=21x$$

Subtract 21x from both sides.

$$66x^{2}-66x-21x=0$$

Combine -66x and -21x to get -87x.

$$66x^{2}-87x=0$$

Divide both sides by 66.

$$\frac{66x^{2}-87x}{66}=\frac{0}{66}$$

Dividing by 66 undoes the multiplication by 66.

$$x^{2}+\left(-\frac{87}{66}\right)x=\frac{0}{66}$$

Reduce the fraction \frac{-87}{66} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{29}{22}x=\frac{0}{66}$$

Divide 0 by 66.

$$x^{2}-\frac{29}{22}x=0$$

Divide -\frac{29}{22}, the coefficient of the x term, by 2 to get -\frac{29}{44}. Then add the square of -\frac{29}{44} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{29}{22}x+\left(-\frac{29}{44}\right)^{2}=\left(-\frac{29}{44}\right)^{2}$$

Square -\frac{29}{44} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{29}{22}x+\frac{841}{1936}=\frac{841}{1936}$$

Factor x^{2}-\frac{29}{22}x+\frac{841}{1936}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{29}{44}\right)^{2}=\frac{841}{1936}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{29}{44}\right)^{2}}=\sqrt{\frac{841}{1936}}$$

Simplify.

$$x-\frac{29}{44}=\frac{29}{44} x-\frac{29}{44}=-\frac{29}{44}$$

Add \frac{29}{44} to both sides of the equation.

$$x=\frac{29}{22} x=0$$