Solve 21left(x-2right)^2-(x-2)=2

$$21 { \left(x-2 \right) }^{ 2 } -(x-2)=2$$

Solve for x

$$x = \frac{12}{7} = 1\frac{5}{7} \approx 1.714285714$$

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

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$21\left(x^{2}-4x+4\right)-\left(x-2\right)=2$$

Use the distributive property to multiply 21 by x^{2}-4x+4.

$$21x^{2}-84x+84-\left(x-2\right)=2$$

To find the opposite of x-2, find the opposite of each term.

$$21x^{2}-84x+84-x+2=2$$

Combine -84x and -x to get -85x.

$$21x^{2}-85x+84+2=2$$

Add 84 and 2 to get 86.

$$21x^{2}-85x+86=2$$

Subtract 2 from both sides.

$$21x^{2}-85x+86-2=0$$

Subtract 2 from 86 to get 84.

$$21x^{2}-85x+84=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, -85 for b, and 84 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-85\right)±\sqrt{\left(-85\right)^{2}-4\times 21\times 84}}{2\times 21}$$

Square -85.

$$x=\frac{-\left(-85\right)±\sqrt{7225-4\times 21\times 84}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-\left(-85\right)±\sqrt{7225-84\times 84}}{2\times 21}$$

Multiply -84 times 84.

$$x=\frac{-\left(-85\right)±\sqrt{7225-7056}}{2\times 21}$$

Add 7225 to -7056.

$$x=\frac{-\left(-85\right)±\sqrt{169}}{2\times 21}$$

Take the square root of 169.

$$x=\frac{-\left(-85\right)±13}{2\times 21}$$

The opposite of -85 is 85.

$$x=\frac{85±13}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{85±13}{42}$$

Now solve the equation x=\frac{85±13}{42} when ± is plus. Add 85 to 13.

$$x=\frac{98}{42}$$

Reduce the fraction \frac{98}{42} to lowest terms by extracting and canceling out 14.

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

Now solve the equation x=\frac{85±13}{42} when ± is minus. Subtract 13 from 85.

$$x=\frac{72}{42}$$

Reduce the fraction \frac{72}{42} to lowest terms by extracting and canceling out 6.

$$x=\frac{12}{7}$$

The equation is now solved.

$$x=\frac{7}{3} x=\frac{12}{7}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$21\left(x^{2}-4x+4\right)-\left(x-2\right)=2$$

Use the distributive property to multiply 21 by x^{2}-4x+4.

$$21x^{2}-84x+84-\left(x-2\right)=2$$

To find the opposite of x-2, find the opposite of each term.

$$21x^{2}-84x+84-x+2=2$$

Combine -84x and -x to get -85x.

$$21x^{2}-85x+84+2=2$$

Add 84 and 2 to get 86.

$$21x^{2}-85x+86=2$$

Subtract 86 from both sides.

$$21x^{2}-85x=2-86$$

Subtract 86 from 2 to get -84.

$$21x^{2}-85x=-84$$

Divide both sides by 21.

$$\frac{21x^{2}-85x}{21}=-\frac{84}{21}$$

Dividing by 21 undoes the multiplication by 21.

$$x^{2}-\frac{85}{21}x=-\frac{84}{21}$$

Divide -84 by 21.

$$x^{2}-\frac{85}{21}x=-4$$

Divide -\frac{85}{21}, the coefficient of the x term, by 2 to get -\frac{85}{42}. Then add the square of -\frac{85}{42} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{85}{21}x+\left(-\frac{85}{42}\right)^{2}=-4+\left(-\frac{85}{42}\right)^{2}$$

Square -\frac{85}{42} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{85}{21}x+\frac{7225}{1764}=-4+\frac{7225}{1764}$$

Add -4 to \frac{7225}{1764}.

$$x^{2}-\frac{85}{21}x+\frac{7225}{1764}=\frac{169}{1764}$$

Factor x^{2}-\frac{85}{21}x+\frac{7225}{1764}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{85}{42}\right)^{2}=\frac{169}{1764}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{85}{42}\right)^{2}}=\sqrt{\frac{169}{1764}}$$

Simplify.

$$x-\frac{85}{42}=\frac{13}{42} x-\frac{85}{42}=-\frac{13}{42}$$

Add \frac{85}{42} to both sides of the equation.

$$x=\frac{7}{3} x=\frac{12}{7}$$