Solve 7(x-2)3/5(x-8)=13/12(x-5)

$$7 ( x - 2 ) \frac { 3 } { 5 } ( x - 8 ) = \frac { 13 } { 12 } ( x - 5 )$$

Solve for x

$$x = \frac{\sqrt{2290369} + 2585}{504} \approx 8.131739109$$

$$x = \frac{2585 - \sqrt{2290369}}{504} \approx 2.126197398$$

Steps Using the Quadratic Formula

Multiply 7 and \frac{3}{5} to get \frac{21}{5}.

$$\frac{21}{5}\left(x-2\right)\left(x-8\right)=\frac{13}{12}\left(x-5\right)$$

Use the distributive property to multiply \frac{21}{5} by x-2.

$$\left(\frac{21}{5}x-\frac{42}{5}\right)\left(x-8\right)=\frac{13}{12}\left(x-5\right)$$

Use the distributive property to multiply \frac{21}{5}x-\frac{42}{5} by x-8 and combine like terms.

$$\frac{21}{5}x^{2}-42x+\frac{336}{5}=\frac{13}{12}\left(x-5\right)$$

Use the distributive property to multiply \frac{13}{12} by x-5.

$$\frac{21}{5}x^{2}-42x+\frac{336}{5}=\frac{13}{12}x-\frac{65}{12}$$

Subtract \frac{13}{12}x from both sides.

$$\frac{21}{5}x^{2}-42x+\frac{336}{5}-\frac{13}{12}x=-\frac{65}{12}$$

Combine -42x and -\frac{13}{12}x to get -\frac{517}{12}x.

$$\frac{21}{5}x^{2}-\frac{517}{12}x+\frac{336}{5}=-\frac{65}{12}$$

Add \frac{65}{12} to both sides.

$$\frac{21}{5}x^{2}-\frac{517}{12}x+\frac{336}{5}+\frac{65}{12}=0$$

Add \frac{336}{5} and \frac{65}{12} to get \frac{4357}{60}.

$$\frac{21}{5}x^{2}-\frac{517}{12}x+\frac{4357}{60}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{21}{5} for a, -\frac{517}{12} for b, and \frac{4357}{60} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{517}{12}\right)±\sqrt{\left(-\frac{517}{12}\right)^{2}-4\times \frac{21}{5}\times \frac{4357}{60}}}{2\times \frac{21}{5}}$$

Square -\frac{517}{12} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{517}{12}\right)±\sqrt{\frac{267289}{144}-4\times \frac{21}{5}\times \frac{4357}{60}}}{2\times \frac{21}{5}}$$

Multiply -4 times \frac{21}{5}.

$$x=\frac{-\left(-\frac{517}{12}\right)±\sqrt{\frac{267289}{144}-\frac{84}{5}\times \frac{4357}{60}}}{2\times \frac{21}{5}}$$

Multiply -\frac{84}{5} times \frac{4357}{60} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{517}{12}\right)±\sqrt{\frac{267289}{144}-\frac{30499}{25}}}{2\times \frac{21}{5}}$$

Add \frac{267289}{144} to -\frac{30499}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{517}{12}\right)±\sqrt{\frac{2290369}{3600}}}{2\times \frac{21}{5}}$$

Take the square root of \frac{2290369}{3600}.

$$x=\frac{-\left(-\frac{517}{12}\right)±\frac{\sqrt{2290369}}{60}}{2\times \frac{21}{5}}$$

The opposite of -\frac{517}{12} is \frac{517}{12}.

$$x=\frac{\frac{517}{12}±\frac{\sqrt{2290369}}{60}}{2\times \frac{21}{5}}$$

Multiply 2 times \frac{21}{5}.

$$x=\frac{\frac{517}{12}±\frac{\sqrt{2290369}}{60}}{\frac{42}{5}}$$

Now solve the equation x=\frac{\frac{517}{12}±\frac{\sqrt{2290369}}{60}}{\frac{42}{5}} when ± is plus. Add \frac{517}{12} to \frac{\sqrt{2290369}}{60}.

$$x=\frac{\frac{\sqrt{2290369}}{60}+\frac{517}{12}}{\frac{42}{5}}$$

Divide \frac{517}{12}+\frac{\sqrt{2290369}}{60} by \frac{42}{5} by multiplying \frac{517}{12}+\frac{\sqrt{2290369}}{60} by the reciprocal of \frac{42}{5}.

$$x=\frac{\sqrt{2290369}+2585}{504}$$

Now solve the equation x=\frac{\frac{517}{12}±\frac{\sqrt{2290369}}{60}}{\frac{42}{5}} when ± is minus. Subtract \frac{\sqrt{2290369}}{60} from \frac{517}{12}.

$$x=\frac{-\frac{\sqrt{2290369}}{60}+\frac{517}{12}}{\frac{42}{5}}$$

Divide \frac{517}{12}-\frac{\sqrt{2290369}}{60} by \frac{42}{5} by multiplying \frac{517}{12}-\frac{\sqrt{2290369}}{60} by the reciprocal of \frac{42}{5}.

$$x=\frac{2585-\sqrt{2290369}}{504}$$

The equation is now solved.

$$x=\frac{\sqrt{2290369}+2585}{504} x=\frac{2585-\sqrt{2290369}}{504}$$

Steps for Completing the Square

Multiply 7 and \frac{3}{5} to get \frac{21}{5}.

$$\frac{21}{5}\left(x-2\right)\left(x-8\right)=\frac{13}{12}\left(x-5\right)$$

Use the distributive property to multiply \frac{21}{5} by x-2.

$$\left(\frac{21}{5}x-\frac{42}{5}\right)\left(x-8\right)=\frac{13}{12}\left(x-5\right)$$

Use the distributive property to multiply \frac{21}{5}x-\frac{42}{5} by x-8 and combine like terms.

$$\frac{21}{5}x^{2}-42x+\frac{336}{5}=\frac{13}{12}\left(x-5\right)$$

Use the distributive property to multiply \frac{13}{12} by x-5.

$$\frac{21}{5}x^{2}-42x+\frac{336}{5}=\frac{13}{12}x-\frac{65}{12}$$

Subtract \frac{13}{12}x from both sides.

$$\frac{21}{5}x^{2}-42x+\frac{336}{5}-\frac{13}{12}x=-\frac{65}{12}$$

Combine -42x and -\frac{13}{12}x to get -\frac{517}{12}x.

$$\frac{21}{5}x^{2}-\frac{517}{12}x+\frac{336}{5}=-\frac{65}{12}$$

Subtract \frac{336}{5} from both sides.

$$\frac{21}{5}x^{2}-\frac{517}{12}x=-\frac{65}{12}-\frac{336}{5}$$

Subtract \frac{336}{5} from -\frac{65}{12} to get -\frac{4357}{60}.

$$\frac{21}{5}x^{2}-\frac{517}{12}x=-\frac{4357}{60}$$

Divide both sides of the equation by \frac{21}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{21}{5}x^{2}-\frac{517}{12}x}{\frac{21}{5}}=-\frac{\frac{4357}{60}}{\frac{21}{5}}$$

Dividing by \frac{21}{5} undoes the multiplication by \frac{21}{5}.

$$x^{2}+\left(-\frac{\frac{517}{12}}{\frac{21}{5}}\right)x=-\frac{\frac{4357}{60}}{\frac{21}{5}}$$

Divide -\frac{517}{12} by \frac{21}{5} by multiplying -\frac{517}{12} by the reciprocal of \frac{21}{5}.

$$x^{2}-\frac{2585}{252}x=-\frac{\frac{4357}{60}}{\frac{21}{5}}$$

Divide -\frac{4357}{60} by \frac{21}{5} by multiplying -\frac{4357}{60} by the reciprocal of \frac{21}{5}.

$$x^{2}-\frac{2585}{252}x=-\frac{4357}{252}$$

Divide -\frac{2585}{252}, the coefficient of the x term, by 2 to get -\frac{2585}{504}. Then add the square of -\frac{2585}{504} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2585}{252}x+\left(-\frac{2585}{504}\right)^{2}=-\frac{4357}{252}+\left(-\frac{2585}{504}\right)^{2}$$

Square -\frac{2585}{504} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2585}{252}x+\frac{6682225}{254016}=-\frac{4357}{252}+\frac{6682225}{254016}$$

Add -\frac{4357}{252} to \frac{6682225}{254016} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2585}{252}x+\frac{6682225}{254016}=\frac{2290369}{254016}$$

Factor x^{2}-\frac{2585}{252}x+\frac{6682225}{254016}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2585}{504}\right)^{2}=\frac{2290369}{254016}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2585}{504}\right)^{2}}=\sqrt{\frac{2290369}{254016}}$$

Simplify.

$$x-\frac{2585}{504}=\frac{\sqrt{2290369}}{504} x-\frac{2585}{504}=-\frac{\sqrt{2290369}}{504}$$

Add \frac{2585}{504} to both sides of the equation.

$$x=\frac{\sqrt{2290369}+2585}{504} x=\frac{2585-\sqrt{2290369}}{504}$$