Solve x+13/4/3=4/3x^2-frac{13/3x}{frac{13}{4}-1}

$$\frac { x + \frac { 13 } { 4 } } { 3 } = \frac { \frac { 4 } { 3 } x ^ { 2 } - \frac { 13 } { 3 } x } { \frac { 13 } { 4 } - 1 }$$

Solve for x

$$x = \frac{\sqrt{5593} + 61}{32} \approx 4.243323825$$

$$x=\frac{61-\sqrt{5593}}{32}\approx -0.430823825$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$x+\frac{13}{4}=\frac{4}{3}\left(\frac{4}{3}x^{2}-\frac{13}{3}x\right)$$

Use the distributive property to multiply \frac{4}{3} by \frac{4}{3}x^{2}-\frac{13}{3}x.

$$x+\frac{13}{4}=\frac{16}{9}x^{2}-\frac{52}{9}x$$

Subtract \frac{16}{9}x^{2} from both sides.

$$x+\frac{13}{4}-\frac{16}{9}x^{2}=-\frac{52}{9}x$$

Add \frac{52}{9}x to both sides.

$$x+\frac{13}{4}-\frac{16}{9}x^{2}+\frac{52}{9}x=0$$

Combine x and \frac{52}{9}x to get \frac{61}{9}x.

$$\frac{61}{9}x+\frac{13}{4}-\frac{16}{9}x^{2}=0$$

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.

$$-\frac{16}{9}x^{2}+\frac{61}{9}x+\frac{13}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{16}{9} for a, \frac{61}{9} for b, and \frac{13}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{61}{9}±\sqrt{\left(\frac{61}{9}\right)^{2}-4\left(-\frac{16}{9}\right)\times \frac{13}{4}}}{2\left(-\frac{16}{9}\right)}$$

Square \frac{61}{9} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{61}{9}±\sqrt{\frac{3721}{81}-4\left(-\frac{16}{9}\right)\times \frac{13}{4}}}{2\left(-\frac{16}{9}\right)}$$

Multiply -4 times -\frac{16}{9}.

$$x=\frac{-\frac{61}{9}±\sqrt{\frac{3721}{81}+\frac{64}{9}\times \frac{13}{4}}}{2\left(-\frac{16}{9}\right)}$$

Multiply \frac{64}{9} times \frac{13}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{61}{9}±\sqrt{\frac{3721}{81}+\frac{208}{9}}}{2\left(-\frac{16}{9}\right)}$$

Add \frac{3721}{81} to \frac{208}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{61}{9}±\sqrt{\frac{5593}{81}}}{2\left(-\frac{16}{9}\right)}$$

Take the square root of \frac{5593}{81}.

$$x=\frac{-\frac{61}{9}±\frac{\sqrt{5593}}{9}}{2\left(-\frac{16}{9}\right)}$$

Multiply 2 times -\frac{16}{9}.

$$x=\frac{-\frac{61}{9}±\frac{\sqrt{5593}}{9}}{-\frac{32}{9}}$$

Now solve the equation x=\frac{-\frac{61}{9}±\frac{\sqrt{5593}}{9}}{-\frac{32}{9}} when ± is plus. Add -\frac{61}{9} to \frac{\sqrt{5593}}{9}.

$$x=\frac{\sqrt{5593}-61}{-\frac{32}{9}\times 9}$$

Divide \frac{-61+\sqrt{5593}}{9} by -\frac{32}{9} by multiplying \frac{-61+\sqrt{5593}}{9} by the reciprocal of -\frac{32}{9}.

$$x=\frac{61-\sqrt{5593}}{32}$$

Now solve the equation x=\frac{-\frac{61}{9}±\frac{\sqrt{5593}}{9}}{-\frac{32}{9}} when ± is minus. Subtract \frac{\sqrt{5593}}{9} from -\frac{61}{9}.

$$x=\frac{-\sqrt{5593}-61}{-\frac{32}{9}\times 9}$$

Divide \frac{-61-\sqrt{5593}}{9} by -\frac{32}{9} by multiplying \frac{-61-\sqrt{5593}}{9} by the reciprocal of -\frac{32}{9}.

$$x=\frac{\sqrt{5593}+61}{32}$$

The equation is now solved.

$$x=\frac{61-\sqrt{5593}}{32} x=\frac{\sqrt{5593}+61}{32}$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$x+\frac{13}{4}=\frac{4}{3}\left(\frac{4}{3}x^{2}-\frac{13}{3}x\right)$$

Use the distributive property to multiply \frac{4}{3} by \frac{4}{3}x^{2}-\frac{13}{3}x.

$$x+\frac{13}{4}=\frac{16}{9}x^{2}-\frac{52}{9}x$$

Subtract \frac{16}{9}x^{2} from both sides.

$$x+\frac{13}{4}-\frac{16}{9}x^{2}=-\frac{52}{9}x$$

Add \frac{52}{9}x to both sides.

$$x+\frac{13}{4}-\frac{16}{9}x^{2}+\frac{52}{9}x=0$$

Combine x and \frac{52}{9}x to get \frac{61}{9}x.

$$\frac{61}{9}x+\frac{13}{4}-\frac{16}{9}x^{2}=0$$

Subtract \frac{13}{4} from both sides. Anything subtracted from zero gives its negation.

$$\frac{61}{9}x-\frac{16}{9}x^{2}=-\frac{13}{4}$$

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.

$$-\frac{16}{9}x^{2}+\frac{61}{9}x=-\frac{13}{4}$$

Divide both sides of the equation by -\frac{16}{9}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{16}{9}x^{2}+\frac{61}{9}x}{-\frac{16}{9}}=-\frac{\frac{13}{4}}{-\frac{16}{9}}$$

Dividing by -\frac{16}{9} undoes the multiplication by -\frac{16}{9}.

$$x^{2}+\frac{\frac{61}{9}}{-\frac{16}{9}}x=-\frac{\frac{13}{4}}{-\frac{16}{9}}$$

Divide \frac{61}{9} by -\frac{16}{9} by multiplying \frac{61}{9} by the reciprocal of -\frac{16}{9}.

$$x^{2}-\frac{61}{16}x=-\frac{\frac{13}{4}}{-\frac{16}{9}}$$

Divide -\frac{13}{4} by -\frac{16}{9} by multiplying -\frac{13}{4} by the reciprocal of -\frac{16}{9}.

$$x^{2}-\frac{61}{16}x=\frac{117}{64}$$

Divide -\frac{61}{16}, the coefficient of the x term, by 2 to get -\frac{61}{32}. Then add the square of -\frac{61}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{61}{16}x+\left(-\frac{61}{32}\right)^{2}=\frac{117}{64}+\left(-\frac{61}{32}\right)^{2}$$

Square -\frac{61}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{61}{16}x+\frac{3721}{1024}=\frac{117}{64}+\frac{3721}{1024}$$

Add \frac{117}{64} to \frac{3721}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{61}{16}x+\frac{3721}{1024}=\frac{5593}{1024}$$

Factor x^{2}-\frac{61}{16}x+\frac{3721}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{61}{32}\right)^{2}=\frac{5593}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{61}{32}\right)^{2}}=\sqrt{\frac{5593}{1024}}$$

Simplify.

$$x-\frac{61}{32}=\frac{\sqrt{5593}}{32} x-\frac{61}{32}=-\frac{\sqrt{5593}}{32}$$

Add \frac{61}{32} to both sides of the equation.

$$x=\frac{\sqrt{5593}+61}{32} x=\frac{61-\sqrt{5593}}{32}$$