Solve x+1/8x-1/6+x+3/5=2

$$\frac { x + 1 } { 8 } \frac { x - 1 } { 6 } + \frac { x + 3 } { 5 } = 2$$

Solve for x

$$x = \frac{\sqrt{2281} - 24}{5} \approx 4.751963149$$

$$x=\frac{-\sqrt{2281}-24}{5}\approx -14.351963149$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 120, the least common multiple of 8,6,5.

$$15\left(x+1\right)\times \frac{x-1}{6}+24\left(x+3\right)=240$$

Express 15\times \frac{x-1}{6} as a single fraction.

$$\frac{15\left(x-1\right)}{6}\left(x+1\right)+24\left(x+3\right)=240$$

Use the distributive property to multiply \frac{15\left(x-1\right)}{6} by x+1.

$$\frac{15\left(x-1\right)}{6}x+\frac{15\left(x-1\right)}{6}+24\left(x+3\right)=240$$

Use the distributive property to multiply 15 by x-1.

$$\frac{15x-15}{6}x+\frac{15\left(x-1\right)}{6}+24\left(x+3\right)=240$$

Express \frac{15x-15}{6}x as a single fraction.

$$\frac{\left(15x-15\right)x}{6}+\frac{15\left(x-1\right)}{6}+24\left(x+3\right)=240$$

Use the distributive property to multiply 15 by x-1.

$$\frac{\left(15x-15\right)x}{6}+\frac{15x-15}{6}+24\left(x+3\right)=240$$

Since \frac{\left(15x-15\right)x}{6} and \frac{15x-15}{6} have the same denominator, add them by adding their numerators.

$$\frac{\left(15x-15\right)x+15x-15}{6}+24\left(x+3\right)=240$$

Do the multiplications in \left(15x-15\right)x+15x-15.

$$\frac{15x^{2}-15x+15x-15}{6}+24\left(x+3\right)=240$$

Combine like terms in 15x^{2}-15x+15x-15.

$$\frac{15x^{2}-15}{6}+24\left(x+3\right)=240$$

Use the distributive property to multiply 24 by x+3.

$$\frac{15x^{2}-15}{6}+24x+72=240$$

Divide each term of 15x^{2}-15 by 6 to get \frac{5}{2}x^{2}-\frac{5}{2}.

$$\frac{5}{2}x^{2}-\frac{5}{2}+24x+72=240$$

Convert 72 to fraction \frac{144}{2}.

$$\frac{5}{2}x^{2}-\frac{5}{2}+24x+\frac{144}{2}=240$$

Since -\frac{5}{2} and \frac{144}{2} have the same denominator, add them by adding their numerators.

$$\frac{5}{2}x^{2}+\frac{-5+144}{2}+24x=240$$

Add -5 and 144 to get 139.

$$\frac{5}{2}x^{2}+\frac{139}{2}+24x=240$$

Subtract 240 from both sides.

$$\frac{5}{2}x^{2}+\frac{139}{2}+24x-240=0$$

Convert 240 to fraction \frac{480}{2}.

$$\frac{5}{2}x^{2}+\frac{139}{2}+24x-\frac{480}{2}=0$$

Since \frac{139}{2} and \frac{480}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5}{2}x^{2}+\frac{139-480}{2}+24x=0$$

Subtract 480 from 139 to get -341.

$$\frac{5}{2}x^{2}-\frac{341}{2}+24x=0$$

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.

$$\frac{5}{2}x^{2}+24x-\frac{341}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{5}{2} for a, 24 for b, and -\frac{341}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times \frac{5}{2}\left(-\frac{341}{2}\right)}}{2\times \frac{5}{2}}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times \frac{5}{2}\left(-\frac{341}{2}\right)}}{2\times \frac{5}{2}}$$

Multiply -4 times \frac{5}{2}.

$$x=\frac{-24±\sqrt{576-10\left(-\frac{341}{2}\right)}}{2\times \frac{5}{2}}$$

Multiply -10 times -\frac{341}{2}.

$$x=\frac{-24±\sqrt{576+1705}}{2\times \frac{5}{2}}$$

Add 576 to 1705.

$$x=\frac{-24±\sqrt{2281}}{2\times \frac{5}{2}}$$

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

$$x=\frac{-24±\sqrt{2281}}{5}$$

Now solve the equation x=\frac{-24±\sqrt{2281}}{5} when ± is plus. Add -24 to \sqrt{2281}.

$$x=\frac{\sqrt{2281}-24}{5}$$

Now solve the equation x=\frac{-24±\sqrt{2281}}{5} when ± is minus. Subtract \sqrt{2281} from -24.

$$x=\frac{-\sqrt{2281}-24}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{2281}-24}{5} x=\frac{-\sqrt{2281}-24}{5}$$

Steps for Completing the Square

Multiply both sides of the equation by 120, the least common multiple of 8,6,5.

$$15\left(x+1\right)\times \frac{x-1}{6}+24\left(x+3\right)=240$$

Express 15\times \frac{x-1}{6} as a single fraction.

$$\frac{15\left(x-1\right)}{6}\left(x+1\right)+24\left(x+3\right)=240$$

Use the distributive property to multiply \frac{15\left(x-1\right)}{6} by x+1.

$$\frac{15\left(x-1\right)}{6}x+\frac{15\left(x-1\right)}{6}+24\left(x+3\right)=240$$

Use the distributive property to multiply 15 by x-1.

$$\frac{15x-15}{6}x+\frac{15\left(x-1\right)}{6}+24\left(x+3\right)=240$$

Express \frac{15x-15}{6}x as a single fraction.

$$\frac{\left(15x-15\right)x}{6}+\frac{15\left(x-1\right)}{6}+24\left(x+3\right)=240$$

Use the distributive property to multiply 15 by x-1.

$$\frac{\left(15x-15\right)x}{6}+\frac{15x-15}{6}+24\left(x+3\right)=240$$

Since \frac{\left(15x-15\right)x}{6} and \frac{15x-15}{6} have the same denominator, add them by adding their numerators.

$$\frac{\left(15x-15\right)x+15x-15}{6}+24\left(x+3\right)=240$$

Do the multiplications in \left(15x-15\right)x+15x-15.

$$\frac{15x^{2}-15x+15x-15}{6}+24\left(x+3\right)=240$$

Combine like terms in 15x^{2}-15x+15x-15.

$$\frac{15x^{2}-15}{6}+24\left(x+3\right)=240$$

Use the distributive property to multiply 24 by x+3.

$$\frac{15x^{2}-15}{6}+24x+72=240$$

Divide each term of 15x^{2}-15 by 6 to get \frac{5}{2}x^{2}-\frac{5}{2}.

$$\frac{5}{2}x^{2}-\frac{5}{2}+24x+72=240$$

Convert 72 to fraction \frac{144}{2}.

$$\frac{5}{2}x^{2}-\frac{5}{2}+24x+\frac{144}{2}=240$$

Since -\frac{5}{2} and \frac{144}{2} have the same denominator, add them by adding their numerators.

$$\frac{5}{2}x^{2}+\frac{-5+144}{2}+24x=240$$

Add -5 and 144 to get 139.

$$\frac{5}{2}x^{2}+\frac{139}{2}+24x=240$$

Subtract \frac{139}{2} from both sides.

$$\frac{5}{2}x^{2}+24x=240-\frac{139}{2}$$

Convert 240 to fraction \frac{480}{2}.

$$\frac{5}{2}x^{2}+24x=\frac{480}{2}-\frac{139}{2}$$

Since \frac{480}{2} and \frac{139}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5}{2}x^{2}+24x=\frac{480-139}{2}$$

Subtract 139 from 480 to get 341.

$$\frac{5}{2}x^{2}+24x=\frac{341}{2}$$

Divide both sides of the equation by \frac{5}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{5}{2}x^{2}+24x}{\frac{5}{2}}=\frac{\frac{341}{2}}{\frac{5}{2}}$$

Dividing by \frac{5}{2} undoes the multiplication by \frac{5}{2}.

$$x^{2}+\frac{24}{\frac{5}{2}}x=\frac{\frac{341}{2}}{\frac{5}{2}}$$

Divide 24 by \frac{5}{2} by multiplying 24 by the reciprocal of \frac{5}{2}.

$$x^{2}+\frac{48}{5}x=\frac{\frac{341}{2}}{\frac{5}{2}}$$

Divide \frac{341}{2} by \frac{5}{2} by multiplying \frac{341}{2} by the reciprocal of \frac{5}{2}.

$$x^{2}+\frac{48}{5}x=\frac{341}{5}$$

Divide \frac{48}{5}, the coefficient of the x term, by 2 to get \frac{24}{5}. Then add the square of \frac{24}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{48}{5}x+\left(\frac{24}{5}\right)^{2}=\frac{341}{5}+\left(\frac{24}{5}\right)^{2}$$

Square \frac{24}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{48}{5}x+\frac{576}{25}=\frac{341}{5}+\frac{576}{25}$$

Add \frac{341}{5} to \frac{576}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{48}{5}x+\frac{576}{25}=\frac{2281}{25}$$

Factor x^{2}+\frac{48}{5}x+\frac{576}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{24}{5}\right)^{2}=\frac{2281}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{24}{5}\right)^{2}}=\sqrt{\frac{2281}{25}}$$

Simplify.

$$x+\frac{24}{5}=\frac{\sqrt{2281}}{5} x+\frac{24}{5}=-\frac{\sqrt{2281}}{5}$$

Subtract \frac{24}{5} from both sides of the equation.

$$x=\frac{\sqrt{2281}-24}{5} x=\frac{-\sqrt{2281}-24}{5}$$