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

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

Solve for x (complex solution)

$$x=\frac{53+\sqrt{11863}i}{56}\approx 0.946428571+1.94495357i$$

$$x=\frac{-\sqrt{11863}i+53}{56}\approx 0.946428571-1.94495357i$$

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 \frac{20}{3}.

$$-\frac{1}{7}x+\frac{3}{7}=\frac{4}{3}x^{2}-\frac{8}{3}x+\frac{20}{3}$$

Subtract \frac{4}{3}x^{2} from both sides.

$$-\frac{1}{7}x+\frac{3}{7}-\frac{4}{3}x^{2}=-\frac{8}{3}x+\frac{20}{3}$$

Add \frac{8}{3}x to both sides.

$$-\frac{1}{7}x+\frac{3}{7}-\frac{4}{3}x^{2}+\frac{8}{3}x=\frac{20}{3}$$

Combine -\frac{1}{7}x and \frac{8}{3}x to get \frac{53}{21}x.

$$\frac{53}{21}x+\frac{3}{7}-\frac{4}{3}x^{2}=\frac{20}{3}$$

Subtract \frac{20}{3} from both sides.

$$\frac{53}{21}x+\frac{3}{7}-\frac{4}{3}x^{2}-\frac{20}{3}=0$$

Subtract \frac{20}{3} from \frac{3}{7} to get -\frac{131}{21}.

$$\frac{53}{21}x-\frac{131}{21}-\frac{4}{3}x^{2}=0$$

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.

$$-\frac{4}{3}x^{2}+\frac{53}{21}x-\frac{131}{21}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{4}{3} for a, \frac{53}{21} for b, and -\frac{131}{21} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{53}{21}±\sqrt{\left(\frac{53}{21}\right)^{2}-4\left(-\frac{4}{3}\right)\left(-\frac{131}{21}\right)}}{2\left(-\frac{4}{3}\right)}$$

Square \frac{53}{21} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{53}{21}±\sqrt{\frac{2809}{441}-4\left(-\frac{4}{3}\right)\left(-\frac{131}{21}\right)}}{2\left(-\frac{4}{3}\right)}$$

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

$$x=\frac{-\frac{53}{21}±\sqrt{\frac{2809}{441}+\frac{16}{3}\left(-\frac{131}{21}\right)}}{2\left(-\frac{4}{3}\right)}$$

Multiply \frac{16}{3} times -\frac{131}{21} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{53}{21}±\sqrt{\frac{2809}{441}-\frac{2096}{63}}}{2\left(-\frac{4}{3}\right)}$$

Add \frac{2809}{441} to -\frac{2096}{63} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{53}{21}±\sqrt{-\frac{11863}{441}}}{2\left(-\frac{4}{3}\right)}$$

Take the square root of -\frac{11863}{441}.

$$x=\frac{-\frac{53}{21}±\frac{\sqrt{11863}i}{21}}{2\left(-\frac{4}{3}\right)}$$

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

$$x=\frac{-\frac{53}{21}±\frac{\sqrt{11863}i}{21}}{-\frac{8}{3}}$$

Now solve the equation x=\frac{-\frac{53}{21}±\frac{\sqrt{11863}i}{21}}{-\frac{8}{3}} when ± is plus. Add -\frac{53}{21} to \frac{i\sqrt{11863}}{21}.

$$x=\frac{-53+\sqrt{11863}i}{-\frac{8}{3}\times 21}$$

Divide \frac{-53+i\sqrt{11863}}{21} by -\frac{8}{3} by multiplying \frac{-53+i\sqrt{11863}}{21} by the reciprocal of -\frac{8}{3}.

$$x=\frac{-\sqrt{11863}i+53}{56}$$

Now solve the equation x=\frac{-\frac{53}{21}±\frac{\sqrt{11863}i}{21}}{-\frac{8}{3}} when ± is minus. Subtract \frac{i\sqrt{11863}}{21} from -\frac{53}{21}.

$$x=\frac{-\sqrt{11863}i-53}{-\frac{8}{3}\times 21}$$

Divide \frac{-53-i\sqrt{11863}}{21} by -\frac{8}{3} by multiplying \frac{-53-i\sqrt{11863}}{21} by the reciprocal of -\frac{8}{3}.

$$x=\frac{53+\sqrt{11863}i}{56}$$

The equation is now solved.

$$x=\frac{-\sqrt{11863}i+53}{56} x=\frac{53+\sqrt{11863}i}{56}$$

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 \frac{20}{3}.

$$-\frac{1}{7}x+\frac{3}{7}=\frac{4}{3}x^{2}-\frac{8}{3}x+\frac{20}{3}$$

Subtract \frac{4}{3}x^{2} from both sides.

$$-\frac{1}{7}x+\frac{3}{7}-\frac{4}{3}x^{2}=-\frac{8}{3}x+\frac{20}{3}$$

Add \frac{8}{3}x to both sides.

$$-\frac{1}{7}x+\frac{3}{7}-\frac{4}{3}x^{2}+\frac{8}{3}x=\frac{20}{3}$$

Combine -\frac{1}{7}x and \frac{8}{3}x to get \frac{53}{21}x.

$$\frac{53}{21}x+\frac{3}{7}-\frac{4}{3}x^{2}=\frac{20}{3}$$

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

$$\frac{53}{21}x-\frac{4}{3}x^{2}=\frac{20}{3}-\frac{3}{7}$$

Subtract \frac{3}{7} from \frac{20}{3} to get \frac{131}{21}.

$$\frac{53}{21}x-\frac{4}{3}x^{2}=\frac{131}{21}$$

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.

$$-\frac{4}{3}x^{2}+\frac{53}{21}x=\frac{131}{21}$$

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

$$\frac{-\frac{4}{3}x^{2}+\frac{53}{21}x}{-\frac{4}{3}}=\frac{\frac{131}{21}}{-\frac{4}{3}}$$

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

$$x^{2}+\frac{\frac{53}{21}}{-\frac{4}{3}}x=\frac{\frac{131}{21}}{-\frac{4}{3}}$$

Divide \frac{53}{21} by -\frac{4}{3} by multiplying \frac{53}{21} by the reciprocal of -\frac{4}{3}.

$$x^{2}-\frac{53}{28}x=\frac{\frac{131}{21}}{-\frac{4}{3}}$$

Divide \frac{131}{21} by -\frac{4}{3} by multiplying \frac{131}{21} by the reciprocal of -\frac{4}{3}.

$$x^{2}-\frac{53}{28}x=-\frac{131}{28}$$

Divide -\frac{53}{28}, the coefficient of the x term, by 2 to get -\frac{53}{56}. Then add the square of -\frac{53}{56} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{53}{28}x+\left(-\frac{53}{56}\right)^{2}=-\frac{131}{28}+\left(-\frac{53}{56}\right)^{2}$$

Square -\frac{53}{56} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{53}{28}x+\frac{2809}{3136}=-\frac{131}{28}+\frac{2809}{3136}$$

Add -\frac{131}{28} to \frac{2809}{3136} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{53}{28}x+\frac{2809}{3136}=-\frac{11863}{3136}$$

Factor x^{2}-\frac{53}{28}x+\frac{2809}{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}.

$$\left(x-\frac{53}{56}\right)^{2}=-\frac{11863}{3136}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{53}{56}\right)^{2}}=\sqrt{-\frac{11863}{3136}}$$

Simplify.

$$x-\frac{53}{56}=\frac{\sqrt{11863}i}{56} x-\frac{53}{56}=-\frac{\sqrt{11863}i}{56}$$

Add \frac{53}{56} to both sides of the equation.

$$x=\frac{53+\sqrt{11863}i}{56} x=\frac{-\sqrt{11863}i+53}{56}$$