Solve 25/9x^2-14/3x+25=-x^2+2x+3-4/3x+4

$$\frac { 25 } { 9 } x ^ { 2 } - \frac { 14 } { 3 } x + 25 = - x ^ { 2 } + 2 x + 3 - \frac { 4 } { 3 } x + 4$$

Solve for x (complex solution)

$$x=\frac{12+3\sqrt{137}i}{17}\approx 0.705882353+2.065535278i$$

$$x=\frac{-3\sqrt{137}i+12}{17}\approx 0.705882353-2.065535278i$$

Steps Using the Quadratic Formula

Combine 2x and -\frac{4}{3}x to get \frac{2}{3}x.

$$\frac{25}{9}x^{2}-\frac{14}{3}x+25=-x^{2}+\frac{2}{3}x+3+4$$

Add 3 and 4 to get 7.

$$\frac{25}{9}x^{2}-\frac{14}{3}x+25=-x^{2}+\frac{2}{3}x+7$$

Add x^{2} to both sides.

$$\frac{25}{9}x^{2}-\frac{14}{3}x+25+x^{2}=\frac{2}{3}x+7$$

Combine \frac{25}{9}x^{2} and x^{2} to get \frac{34}{9}x^{2}.

$$\frac{34}{9}x^{2}-\frac{14}{3}x+25=\frac{2}{3}x+7$$

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

$$\frac{34}{9}x^{2}-\frac{14}{3}x+25-\frac{2}{3}x=7$$

Combine -\frac{14}{3}x and -\frac{2}{3}x to get -\frac{16}{3}x.

$$\frac{34}{9}x^{2}-\frac{16}{3}x+25=7$$

Subtract 7 from both sides.

$$\frac{34}{9}x^{2}-\frac{16}{3}x+25-7=0$$

Subtract 7 from 25 to get 18.

$$\frac{34}{9}x^{2}-\frac{16}{3}x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{34}{9} for a, -\frac{16}{3} for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{16}{3}\right)±\sqrt{\left(-\frac{16}{3}\right)^{2}-4\times \frac{34}{9}\times 18}}{2\times \frac{34}{9}}$$

Square -\frac{16}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{16}{3}\right)±\sqrt{\frac{256}{9}-4\times \frac{34}{9}\times 18}}{2\times \frac{34}{9}}$$

Multiply -4 times \frac{34}{9}.

$$x=\frac{-\left(-\frac{16}{3}\right)±\sqrt{\frac{256}{9}-\frac{136}{9}\times 18}}{2\times \frac{34}{9}}$$

Multiply -\frac{136}{9} times 18.

$$x=\frac{-\left(-\frac{16}{3}\right)±\sqrt{\frac{256}{9}-272}}{2\times \frac{34}{9}}$$

Add \frac{256}{9} to -272.

$$x=\frac{-\left(-\frac{16}{3}\right)±\sqrt{-\frac{2192}{9}}}{2\times \frac{34}{9}}$$

Take the square root of -\frac{2192}{9}.

$$x=\frac{-\left(-\frac{16}{3}\right)±\frac{4\sqrt{137}i}{3}}{2\times \frac{34}{9}}$$

The opposite of -\frac{16}{3} is \frac{16}{3}.

$$x=\frac{\frac{16}{3}±\frac{4\sqrt{137}i}{3}}{2\times \frac{34}{9}}$$

Multiply 2 times \frac{34}{9}.

$$x=\frac{\frac{16}{3}±\frac{4\sqrt{137}i}{3}}{\frac{68}{9}}$$

Now solve the equation x=\frac{\frac{16}{3}±\frac{4\sqrt{137}i}{3}}{\frac{68}{9}} when ± is plus. Add \frac{16}{3} to \frac{4i\sqrt{137}}{3}.

$$x=\frac{16+4\sqrt{137}i}{3\times \frac{68}{9}}$$

Divide \frac{16+4i\sqrt{137}}{3} by \frac{68}{9} by multiplying \frac{16+4i\sqrt{137}}{3} by the reciprocal of \frac{68}{9}.

$$x=\frac{12+3\sqrt{137}i}{17}$$

Now solve the equation x=\frac{\frac{16}{3}±\frac{4\sqrt{137}i}{3}}{\frac{68}{9}} when ± is minus. Subtract \frac{4i\sqrt{137}}{3} from \frac{16}{3}.

$$x=\frac{-4\sqrt{137}i+16}{3\times \frac{68}{9}}$$

Divide \frac{16-4i\sqrt{137}}{3} by \frac{68}{9} by multiplying \frac{16-4i\sqrt{137}}{3} by the reciprocal of \frac{68}{9}.

$$x=\frac{-3\sqrt{137}i+12}{17}$$

The equation is now solved.

$$x=\frac{12+3\sqrt{137}i}{17} x=\frac{-3\sqrt{137}i+12}{17}$$

Steps for Completing the Square

Combine 2x and -\frac{4}{3}x to get \frac{2}{3}x.

$$\frac{25}{9}x^{2}-\frac{14}{3}x+25=-x^{2}+\frac{2}{3}x+3+4$$

Add 3 and 4 to get 7.

$$\frac{25}{9}x^{2}-\frac{14}{3}x+25=-x^{2}+\frac{2}{3}x+7$$

Add x^{2} to both sides.

$$\frac{25}{9}x^{2}-\frac{14}{3}x+25+x^{2}=\frac{2}{3}x+7$$

Combine \frac{25}{9}x^{2} and x^{2} to get \frac{34}{9}x^{2}.

$$\frac{34}{9}x^{2}-\frac{14}{3}x+25=\frac{2}{3}x+7$$

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

$$\frac{34}{9}x^{2}-\frac{14}{3}x+25-\frac{2}{3}x=7$$

Combine -\frac{14}{3}x and -\frac{2}{3}x to get -\frac{16}{3}x.

$$\frac{34}{9}x^{2}-\frac{16}{3}x+25=7$$

Subtract 25 from both sides.

$$\frac{34}{9}x^{2}-\frac{16}{3}x=7-25$$

Subtract 25 from 7 to get -18.

$$\frac{34}{9}x^{2}-\frac{16}{3}x=-18$$

Divide both sides of the equation by \frac{34}{9}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{34}{9}x^{2}-\frac{16}{3}x}{\frac{34}{9}}=-\frac{18}{\frac{34}{9}}$$

Dividing by \frac{34}{9} undoes the multiplication by \frac{34}{9}.

$$x^{2}+\left(-\frac{\frac{16}{3}}{\frac{34}{9}}\right)x=-\frac{18}{\frac{34}{9}}$$

Divide -\frac{16}{3} by \frac{34}{9} by multiplying -\frac{16}{3} by the reciprocal of \frac{34}{9}.

$$x^{2}-\frac{24}{17}x=-\frac{18}{\frac{34}{9}}$$

Divide -18 by \frac{34}{9} by multiplying -18 by the reciprocal of \frac{34}{9}.

$$x^{2}-\frac{24}{17}x=-\frac{81}{17}$$

Divide -\frac{24}{17}, the coefficient of the x term, by 2 to get -\frac{12}{17}. Then add the square of -\frac{12}{17} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{24}{17}x+\left(-\frac{12}{17}\right)^{2}=-\frac{81}{17}+\left(-\frac{12}{17}\right)^{2}$$

Square -\frac{12}{17} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{24}{17}x+\frac{144}{289}=-\frac{81}{17}+\frac{144}{289}$$

Add -\frac{81}{17} to \frac{144}{289} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{24}{17}x+\frac{144}{289}=-\frac{1233}{289}$$

Factor x^{2}-\frac{24}{17}x+\frac{144}{289}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{12}{17}\right)^{2}=-\frac{1233}{289}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{12}{17}\right)^{2}}=\sqrt{-\frac{1233}{289}}$$

Simplify.

$$x-\frac{12}{17}=\frac{3\sqrt{137}i}{17} x-\frac{12}{17}=-\frac{3\sqrt{137}i}{17}$$

Add \frac{12}{17} to both sides of the equation.

$$x=\frac{12+3\sqrt{137}i}{17} x=\frac{-3\sqrt{137}i+12}{17}$$