Solve x3/4x+4/5x=2

$$x \frac { 3 } { 4 } x + \frac { 4 } { 5 } x = 2$$

Solve for x

$$x = \frac{2 \sqrt{166} - 8}{15} \approx 1.184546497$$

$$x=\frac{-2\sqrt{166}-8}{15}\approx -2.251213164$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$x^{2}\times \frac{3}{4}+\frac{4}{5}x=2$$

Subtract 2 from both sides.

$$x^{2}\times \frac{3}{4}+\frac{4}{5}x-2=0$$

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.

$$\frac{3}{4}x^{2}+\frac{4}{5}x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{3}{4} for a, \frac{4}{5} for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{4}{5}±\sqrt{\left(\frac{4}{5}\right)^{2}-4\times \frac{3}{4}\left(-2\right)}}{2\times \frac{3}{4}}$$

Square \frac{4}{5} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{4}{5}±\sqrt{\frac{16}{25}-4\times \frac{3}{4}\left(-2\right)}}{2\times \frac{3}{4}}$$

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

$$x=\frac{-\frac{4}{5}±\sqrt{\frac{16}{25}-3\left(-2\right)}}{2\times \frac{3}{4}}$$

Multiply -3 times -2.

$$x=\frac{-\frac{4}{5}±\sqrt{\frac{16}{25}+6}}{2\times \frac{3}{4}}$$

Add \frac{16}{25} to 6.

$$x=\frac{-\frac{4}{5}±\sqrt{\frac{166}{25}}}{2\times \frac{3}{4}}$$

Take the square root of \frac{166}{25}.

$$x=\frac{-\frac{4}{5}±\frac{\sqrt{166}}{5}}{2\times \frac{3}{4}}$$

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

$$x=\frac{-\frac{4}{5}±\frac{\sqrt{166}}{5}}{\frac{3}{2}}$$

Now solve the equation x=\frac{-\frac{4}{5}±\frac{\sqrt{166}}{5}}{\frac{3}{2}} when ± is plus. Add -\frac{4}{5} to \frac{\sqrt{166}}{5}.

$$x=\frac{\sqrt{166}-4}{\frac{3}{2}\times 5}$$

Divide \frac{-4+\sqrt{166}}{5} by \frac{3}{2} by multiplying \frac{-4+\sqrt{166}}{5} by the reciprocal of \frac{3}{2}.

$$x=\frac{2\sqrt{166}-8}{15}$$

Now solve the equation x=\frac{-\frac{4}{5}±\frac{\sqrt{166}}{5}}{\frac{3}{2}} when ± is minus. Subtract \frac{\sqrt{166}}{5} from -\frac{4}{5}.

$$x=\frac{-\sqrt{166}-4}{\frac{3}{2}\times 5}$$

Divide \frac{-4-\sqrt{166}}{5} by \frac{3}{2} by multiplying \frac{-4-\sqrt{166}}{5} by the reciprocal of \frac{3}{2}.

$$x=\frac{-2\sqrt{166}-8}{15}$$

The equation is now solved.

$$x=\frac{2\sqrt{166}-8}{15} x=\frac{-2\sqrt{166}-8}{15}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$x^{2}\times \frac{3}{4}+\frac{4}{5}x=2$$

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.

$$\frac{3}{4}x^{2}+\frac{4}{5}x=2$$

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

$$\frac{\frac{3}{4}x^{2}+\frac{4}{5}x}{\frac{3}{4}}=\frac{2}{\frac{3}{4}}$$

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

$$x^{2}+\frac{\frac{4}{5}}{\frac{3}{4}}x=\frac{2}{\frac{3}{4}}$$

Divide \frac{4}{5} by \frac{3}{4} by multiplying \frac{4}{5} by the reciprocal of \frac{3}{4}.

$$x^{2}+\frac{16}{15}x=\frac{2}{\frac{3}{4}}$$

Divide 2 by \frac{3}{4} by multiplying 2 by the reciprocal of \frac{3}{4}.

$$x^{2}+\frac{16}{15}x=\frac{8}{3}$$

Divide \frac{16}{15}, the coefficient of the x term, by 2 to get \frac{8}{15}. Then add the square of \frac{8}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{15}x+\left(\frac{8}{15}\right)^{2}=\frac{8}{3}+\left(\frac{8}{15}\right)^{2}$$

Square \frac{8}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{15}x+\frac{64}{225}=\frac{8}{3}+\frac{64}{225}$$

Add \frac{8}{3} to \frac{64}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{16}{15}x+\frac{64}{225}=\frac{664}{225}$$

Factor x^{2}+\frac{16}{15}x+\frac{64}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{15}\right)^{2}=\frac{664}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{15}\right)^{2}}=\sqrt{\frac{664}{225}}$$

Simplify.

$$x+\frac{8}{15}=\frac{2\sqrt{166}}{15} x+\frac{8}{15}=-\frac{2\sqrt{166}}{15}$$

Subtract \frac{8}{15} from both sides of the equation.

$$x=\frac{2\sqrt{166}-8}{15} x=\frac{-2\sqrt{166}-8}{15}$$