Solve x-2/6*x+1/3-x+3/5=5/2

$$\frac { x - 2 } { 6 } \cdot \frac { x + 1 } { 3 } - \frac { x + 3 } { 5 } = \frac { 5 } { 2 }$$

Solve for x

$$x = \frac{3 \sqrt{701} + 23}{10} \approx 10.242921377$$

$$x=\frac{23-3\sqrt{701}}{10}\approx -5.642921377$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 30, the least common multiple of 6,3,5,2.

$$5\left(x-2\right)\times \frac{x+1}{3}-6\left(x+3\right)=75$$

Express 5\times \frac{x+1}{3} as a single fraction.

$$\frac{5\left(x+1\right)}{3}\left(x-2\right)-6\left(x+3\right)=75$$

Use the distributive property to multiply \frac{5\left(x+1\right)}{3} by x-2.

$$\frac{5\left(x+1\right)}{3}x-2\times \frac{5\left(x+1\right)}{3}-6\left(x+3\right)=75$$

Use the distributive property to multiply 5 by x+1.

$$\frac{5x+5}{3}x-2\times \frac{5\left(x+1\right)}{3}-6\left(x+3\right)=75$$

Express \frac{5x+5}{3}x as a single fraction.

$$\frac{\left(5x+5\right)x}{3}-2\times \frac{5\left(x+1\right)}{3}-6\left(x+3\right)=75$$

Use the distributive property to multiply 5 by x+1.

$$\frac{\left(5x+5\right)x}{3}-2\times \frac{5x+5}{3}-6\left(x+3\right)=75$$

Express -2\times \frac{5x+5}{3} as a single fraction.

$$\frac{\left(5x+5\right)x}{3}+\frac{-2\left(5x+5\right)}{3}-6\left(x+3\right)=75$$

Since \frac{\left(5x+5\right)x}{3} and \frac{-2\left(5x+5\right)}{3} have the same denominator, add them by adding their numerators.

$$\frac{\left(5x+5\right)x-2\left(5x+5\right)}{3}-6\left(x+3\right)=75$$

Do the multiplications in \left(5x+5\right)x-2\left(5x+5\right).

$$\frac{5x^{2}+5x-10x-10}{3}-6\left(x+3\right)=75$$

Combine like terms in 5x^{2}+5x-10x-10.

$$\frac{5x^{2}-5x-10}{3}-6\left(x+3\right)=75$$

Use the distributive property to multiply -6 by x+3.

$$\frac{5x^{2}-5x-10}{3}-6x-18=75$$

Divide each term of 5x^{2}-5x-10 by 3 to get \frac{5}{3}x^{2}-\frac{5}{3}x-\frac{10}{3}.

$$\frac{5}{3}x^{2}-\frac{5}{3}x-\frac{10}{3}-6x-18=75$$

Combine -\frac{5}{3}x and -6x to get -\frac{23}{3}x.

$$\frac{5}{3}x^{2}-\frac{23}{3}x-\frac{10}{3}-18=75$$

Convert 18 to fraction \frac{54}{3}.

$$\frac{5}{3}x^{2}-\frac{23}{3}x-\frac{10}{3}-\frac{54}{3}=75$$

Since -\frac{10}{3} and \frac{54}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5}{3}x^{2}-\frac{23}{3}x+\frac{-10-54}{3}=75$$

Subtract 54 from -10 to get -64.

$$\frac{5}{3}x^{2}-\frac{23}{3}x-\frac{64}{3}=75$$

Subtract 75 from both sides.

$$\frac{5}{3}x^{2}-\frac{23}{3}x-\frac{64}{3}-75=0$$

Convert 75 to fraction \frac{225}{3}.

$$\frac{5}{3}x^{2}-\frac{23}{3}x-\frac{64}{3}-\frac{225}{3}=0$$

Since -\frac{64}{3} and \frac{225}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5}{3}x^{2}-\frac{23}{3}x+\frac{-64-225}{3}=0$$

Subtract 225 from -64 to get -289.

$$\frac{5}{3}x^{2}-\frac{23}{3}x-\frac{289}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{5}{3} for a, -\frac{23}{3} for b, and -\frac{289}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{23}{3}\right)±\sqrt{\left(-\frac{23}{3}\right)^{2}-4\times \frac{5}{3}\left(-\frac{289}{3}\right)}}{2\times \frac{5}{3}}$$

Square -\frac{23}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{23}{3}\right)±\sqrt{\frac{529}{9}-4\times \frac{5}{3}\left(-\frac{289}{3}\right)}}{2\times \frac{5}{3}}$$

Multiply -4 times \frac{5}{3}.

$$x=\frac{-\left(-\frac{23}{3}\right)±\sqrt{\frac{529}{9}-\frac{20}{3}\left(-\frac{289}{3}\right)}}{2\times \frac{5}{3}}$$

Multiply -\frac{20}{3} times -\frac{289}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{23}{3}\right)±\sqrt{\frac{529+5780}{9}}}{2\times \frac{5}{3}}$$

Add \frac{529}{9} to \frac{5780}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{23}{3}\right)±\sqrt{701}}{2\times \frac{5}{3}}$$

The opposite of -\frac{23}{3} is \frac{23}{3}.

$$x=\frac{\frac{23}{3}±\sqrt{701}}{2\times \frac{5}{3}}$$

Multiply 2 times \frac{5}{3}.

$$x=\frac{\frac{23}{3}±\sqrt{701}}{\frac{10}{3}}$$

Now solve the equation x=\frac{\frac{23}{3}±\sqrt{701}}{\frac{10}{3}} when ± is plus. Add \frac{23}{3} to \sqrt{701}.

$$x=\frac{\sqrt{701}+\frac{23}{3}}{\frac{10}{3}}$$

Divide \frac{23}{3}+\sqrt{701} by \frac{10}{3} by multiplying \frac{23}{3}+\sqrt{701} by the reciprocal of \frac{10}{3}.

$$x=\frac{3\sqrt{701}+23}{10}$$

Now solve the equation x=\frac{\frac{23}{3}±\sqrt{701}}{\frac{10}{3}} when ± is minus. Subtract \sqrt{701} from \frac{23}{3}.

$$x=\frac{\frac{23}{3}-\sqrt{701}}{\frac{10}{3}}$$

Divide \frac{23}{3}-\sqrt{701} by \frac{10}{3} by multiplying \frac{23}{3}-\sqrt{701} by the reciprocal of \frac{10}{3}.

$$x=\frac{23-3\sqrt{701}}{10}$$

The equation is now solved.

$$x=\frac{3\sqrt{701}+23}{10} x=\frac{23-3\sqrt{701}}{10}$$

Steps for Completing the Square

Multiply both sides of the equation by 30, the least common multiple of 6,3,5,2.

$$5\left(x-2\right)\times \frac{x+1}{3}-6\left(x+3\right)=75$$

Express 5\times \frac{x+1}{3} as a single fraction.

$$\frac{5\left(x+1\right)}{3}\left(x-2\right)-6\left(x+3\right)=75$$

Use the distributive property to multiply \frac{5\left(x+1\right)}{3} by x-2.

$$\frac{5\left(x+1\right)}{3}x-2\times \frac{5\left(x+1\right)}{3}-6\left(x+3\right)=75$$

Use the distributive property to multiply 5 by x+1.

$$\frac{5x+5}{3}x-2\times \frac{5\left(x+1\right)}{3}-6\left(x+3\right)=75$$

Express \frac{5x+5}{3}x as a single fraction.

$$\frac{\left(5x+5\right)x}{3}-2\times \frac{5\left(x+1\right)}{3}-6\left(x+3\right)=75$$

Use the distributive property to multiply 5 by x+1.

$$\frac{\left(5x+5\right)x}{3}-2\times \frac{5x+5}{3}-6\left(x+3\right)=75$$

Express -2\times \frac{5x+5}{3} as a single fraction.

$$\frac{\left(5x+5\right)x}{3}+\frac{-2\left(5x+5\right)}{3}-6\left(x+3\right)=75$$

Since \frac{\left(5x+5\right)x}{3} and \frac{-2\left(5x+5\right)}{3} have the same denominator, add them by adding their numerators.

$$\frac{\left(5x+5\right)x-2\left(5x+5\right)}{3}-6\left(x+3\right)=75$$

Do the multiplications in \left(5x+5\right)x-2\left(5x+5\right).

$$\frac{5x^{2}+5x-10x-10}{3}-6\left(x+3\right)=75$$

Combine like terms in 5x^{2}+5x-10x-10.

$$\frac{5x^{2}-5x-10}{3}-6\left(x+3\right)=75$$

Use the distributive property to multiply -6 by x+3.

$$\frac{5x^{2}-5x-10}{3}-6x-18=75$$

Divide each term of 5x^{2}-5x-10 by 3 to get \frac{5}{3}x^{2}-\frac{5}{3}x-\frac{10}{3}.

$$\frac{5}{3}x^{2}-\frac{5}{3}x-\frac{10}{3}-6x-18=75$$

Combine -\frac{5}{3}x and -6x to get -\frac{23}{3}x.

$$\frac{5}{3}x^{2}-\frac{23}{3}x-\frac{10}{3}-18=75$$

Convert 18 to fraction \frac{54}{3}.

$$\frac{5}{3}x^{2}-\frac{23}{3}x-\frac{10}{3}-\frac{54}{3}=75$$

Since -\frac{10}{3} and \frac{54}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5}{3}x^{2}-\frac{23}{3}x+\frac{-10-54}{3}=75$$

Subtract 54 from -10 to get -64.

$$\frac{5}{3}x^{2}-\frac{23}{3}x-\frac{64}{3}=75$$

Add \frac{64}{3} to both sides.

$$\frac{5}{3}x^{2}-\frac{23}{3}x=75+\frac{64}{3}$$

Convert 75 to fraction \frac{225}{3}.

$$\frac{5}{3}x^{2}-\frac{23}{3}x=\frac{225}{3}+\frac{64}{3}$$

Since \frac{225}{3} and \frac{64}{3} have the same denominator, add them by adding their numerators.

$$\frac{5}{3}x^{2}-\frac{23}{3}x=\frac{225+64}{3}$$

Add 225 and 64 to get 289.

$$\frac{5}{3}x^{2}-\frac{23}{3}x=\frac{289}{3}$$

Divide both sides of the equation by \frac{5}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{5}{3}x^{2}-\frac{23}{3}x}{\frac{5}{3}}=\frac{\frac{289}{3}}{\frac{5}{3}}$$

Dividing by \frac{5}{3} undoes the multiplication by \frac{5}{3}.

$$x^{2}+\left(-\frac{\frac{23}{3}}{\frac{5}{3}}\right)x=\frac{\frac{289}{3}}{\frac{5}{3}}$$

Divide -\frac{23}{3} by \frac{5}{3} by multiplying -\frac{23}{3} by the reciprocal of \frac{5}{3}.

$$x^{2}-\frac{23}{5}x=\frac{\frac{289}{3}}{\frac{5}{3}}$$

Divide \frac{289}{3} by \frac{5}{3} by multiplying \frac{289}{3} by the reciprocal of \frac{5}{3}.

$$x^{2}-\frac{23}{5}x=\frac{289}{5}$$

Divide -\frac{23}{5}, the coefficient of the x term, by 2 to get -\frac{23}{10}. Then add the square of -\frac{23}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{23}{5}x+\left(-\frac{23}{10}\right)^{2}=\frac{289}{5}+\left(-\frac{23}{10}\right)^{2}$$

Square -\frac{23}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{23}{5}x+\frac{529}{100}=\frac{289}{5}+\frac{529}{100}$$

Add \frac{289}{5} to \frac{529}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{23}{5}x+\frac{529}{100}=\frac{6309}{100}$$

Factor x^{2}-\frac{23}{5}x+\frac{529}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{23}{10}\right)^{2}=\frac{6309}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{10}\right)^{2}}=\sqrt{\frac{6309}{100}}$$

Simplify.

$$x-\frac{23}{10}=\frac{3\sqrt{701}}{10} x-\frac{23}{10}=-\frac{3\sqrt{701}}{10}$$

Add \frac{23}{10} to both sides of the equation.

$$x=\frac{3\sqrt{701}+23}{10} x=\frac{23-3\sqrt{701}}{10}$$