Solve -1/7x+3/7=-4/3left(x-1right)^2+16/3

$$- \frac{ 1 }{ 7 } x+ \frac{ 3 }{ 7 } = - \frac{ 4 }{ 3 } { \left(x-1 \right) }^{ 2 } + \frac{ 16 }{ 3 }$$

Solve for x

$$x=-\frac{25}{28}\approx -0.892857143$$

$$x=3$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$-\frac{1}{7}x+\frac{3}{7}=-\frac{4}{3}\left(x^{2}-2x+1\right)+\frac{16}{3}$$

Use the distributive property to multiply -\frac{4}{3} by x^{2}-2x+1.

$$-\frac{1}{7}x+\frac{3}{7}=-\frac{4}{3}x^{2}+\frac{8}{3}x-\frac{4}{3}+\frac{16}{3}$$

Add -\frac{4}{3} and \frac{16}{3} to get 4.

$$-\frac{1}{7}x+\frac{3}{7}=-\frac{4}{3}x^{2}+\frac{8}{3}x+4$$

Add \frac{4}{3}x^{2} to both sides.

$$-\frac{1}{7}x+\frac{3}{7}+\frac{4}{3}x^{2}=\frac{8}{3}x+4$$

Subtract \frac{8}{3}x from both sides.

$$-\frac{1}{7}x+\frac{3}{7}+\frac{4}{3}x^{2}-\frac{8}{3}x=4$$

Combine -\frac{1}{7}x and -\frac{8}{3}x to get -\frac{59}{21}x.

$$-\frac{59}{21}x+\frac{3}{7}+\frac{4}{3}x^{2}=4$$

Subtract 4 from both sides.

$$-\frac{59}{21}x+\frac{3}{7}+\frac{4}{3}x^{2}-4=0$$

Subtract 4 from \frac{3}{7} to get -\frac{25}{7}.

$$-\frac{59}{21}x-\frac{25}{7}+\frac{4}{3}x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$\frac{4}{3}x^{2}-\frac{59}{21}x-\frac{25}{7}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{4}{3} for a, -\frac{59}{21} for b, and -\frac{25}{7} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{59}{21}\right)±\sqrt{\left(-\frac{59}{21}\right)^{2}-4\times \frac{4}{3}\left(-\frac{25}{7}\right)}}{2\times \frac{4}{3}}$$

Square -\frac{59}{21} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{59}{21}\right)±\sqrt{\frac{3481}{441}-4\times \frac{4}{3}\left(-\frac{25}{7}\right)}}{2\times \frac{4}{3}}$$

Multiply -4 times \frac{4}{3}.

$$x=\frac{-\left(-\frac{59}{21}\right)±\sqrt{\frac{3481}{441}-\frac{16}{3}\left(-\frac{25}{7}\right)}}{2\times \frac{4}{3}}$$

Multiply -\frac{16}{3} times -\frac{25}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{59}{21}\right)±\sqrt{\frac{3481}{441}+\frac{400}{21}}}{2\times \frac{4}{3}}$$

Add \frac{3481}{441} to \frac{400}{21} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{59}{21}\right)±\sqrt{\frac{11881}{441}}}{2\times \frac{4}{3}}$$

Take the square root of \frac{11881}{441}.

$$x=\frac{-\left(-\frac{59}{21}\right)±\frac{109}{21}}{2\times \frac{4}{3}}$$

The opposite of -\frac{59}{21} is \frac{59}{21}.

$$x=\frac{\frac{59}{21}±\frac{109}{21}}{2\times \frac{4}{3}}$$

Multiply 2 times \frac{4}{3}.

$$x=\frac{\frac{59}{21}±\frac{109}{21}}{\frac{8}{3}}$$

Now solve the equation x=\frac{\frac{59}{21}±\frac{109}{21}}{\frac{8}{3}} when ± is plus. Add \frac{59}{21} to \frac{109}{21} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{8}{\frac{8}{3}}$$

Divide 8 by \frac{8}{3} by multiplying 8 by the reciprocal of \frac{8}{3}.

$$x=3$$

Now solve the equation x=\frac{\frac{59}{21}±\frac{109}{21}}{\frac{8}{3}} when ± is minus. Subtract \frac{109}{21} from \frac{59}{21} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{50}{21}}{\frac{8}{3}}$$

Divide -\frac{50}{21} by \frac{8}{3} by multiplying -\frac{50}{21} by the reciprocal of \frac{8}{3}.

$$x=-\frac{25}{28}$$

The equation is now solved.

$$x=3 x=-\frac{25}{28}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$-\frac{1}{7}x+\frac{3}{7}=-\frac{4}{3}\left(x^{2}-2x+1\right)+\frac{16}{3}$$

Use the distributive property to multiply -\frac{4}{3} by x^{2}-2x+1.

$$-\frac{1}{7}x+\frac{3}{7}=-\frac{4}{3}x^{2}+\frac{8}{3}x-\frac{4}{3}+\frac{16}{3}$$

Add -\frac{4}{3} and \frac{16}{3} to get 4.

$$-\frac{1}{7}x+\frac{3}{7}=-\frac{4}{3}x^{2}+\frac{8}{3}x+4$$

Add \frac{4}{3}x^{2} to both sides.

$$-\frac{1}{7}x+\frac{3}{7}+\frac{4}{3}x^{2}=\frac{8}{3}x+4$$

Subtract \frac{8}{3}x from both sides.

$$-\frac{1}{7}x+\frac{3}{7}+\frac{4}{3}x^{2}-\frac{8}{3}x=4$$

Combine -\frac{1}{7}x and -\frac{8}{3}x to get -\frac{59}{21}x.

$$-\frac{59}{21}x+\frac{3}{7}+\frac{4}{3}x^{2}=4$$

Subtract \frac{3}{7} from both sides.

$$-\frac{59}{21}x+\frac{4}{3}x^{2}=4-\frac{3}{7}$$

Subtract \frac{3}{7} from 4 to get \frac{25}{7}.

$$-\frac{59}{21}x+\frac{4}{3}x^{2}=\frac{25}{7}$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$\frac{4}{3}x^{2}-\frac{59}{21}x=\frac{25}{7}$$

Divide both sides of the equation by \frac{4}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{4}{3}x^{2}-\frac{59}{21}x}{\frac{4}{3}}=\frac{\frac{25}{7}}{\frac{4}{3}}$$

Dividing by \frac{4}{3} undoes the multiplication by \frac{4}{3}.

$$x^{2}+\left(-\frac{\frac{59}{21}}{\frac{4}{3}}\right)x=\frac{\frac{25}{7}}{\frac{4}{3}}$$

Divide -\frac{59}{21} by \frac{4}{3} by multiplying -\frac{59}{21} by the reciprocal of \frac{4}{3}.

$$x^{2}-\frac{59}{28}x=\frac{\frac{25}{7}}{\frac{4}{3}}$$

Divide \frac{25}{7} by \frac{4}{3} by multiplying \frac{25}{7} by the reciprocal of \frac{4}{3}.

$$x^{2}-\frac{59}{28}x=\frac{75}{28}$$

Divide -\frac{59}{28}, the coefficient of the x term, by 2 to get -\frac{59}{56}. Then add the square of -\frac{59}{56} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{59}{28}x+\left(-\frac{59}{56}\right)^{2}=\frac{75}{28}+\left(-\frac{59}{56}\right)^{2}$$

Square -\frac{59}{56} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{59}{28}x+\frac{3481}{3136}=\frac{75}{28}+\frac{3481}{3136}$$

Add \frac{75}{28} to \frac{3481}{3136} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{59}{28}x+\frac{3481}{3136}=\frac{11881}{3136}$$

Factor x^{2}-\frac{59}{28}x+\frac{3481}{3136}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{59}{56}\right)^{2}=\frac{11881}{3136}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{59}{56}\right)^{2}}=\sqrt{\frac{11881}{3136}}$$

Simplify.

$$x-\frac{59}{56}=\frac{109}{56} x-\frac{59}{56}=-\frac{109}{56}$$

Add \frac{59}{56} to both sides of the equation.

$$x=3 x=-\frac{25}{28}$$