Solve (X+3)/(2X-1)-(5X-1)/(4X+7)=0

$$( X + 3 ) / ( 2 X - 1 ) - ( 5 X - 1 ) / ( 4 X + 7 ) = 0$$

Solve for X

$$X=5$$

$$X=-\frac{2}{3}\approx -0.666666667$$

Steps Using Factoring By Grouping

Variable X cannot be equal to any of the values -\frac{7}{4},\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2X-1\right)\left(4X+7\right), the least common multiple of 2X-1,4X+7.

$$\left(4X+7\right)\left(X+3\right)-\left(2X-1\right)\left(5X-1\right)=0$$

Use the distributive property to multiply 4X+7 by X+3 and combine like terms.

$$4X^{2}+19X+21-\left(2X-1\right)\left(5X-1\right)=0$$

Use the distributive property to multiply 2X-1 by 5X-1 and combine like terms.

$$4X^{2}+19X+21-\left(10X^{2}-7X+1\right)=0$$

To find the opposite of 10X^{2}-7X+1, find the opposite of each term.

$$4X^{2}+19X+21-10X^{2}+7X-1=0$$

Combine 4X^{2} and -10X^{2} to get -6X^{2}.

$$-6X^{2}+19X+21+7X-1=0$$

Combine 19X and 7X to get 26X.

$$-6X^{2}+26X+21-1=0$$

Subtract 1 from 21 to get 20.

$$-6X^{2}+26X+20=0$$

Divide both sides by 2.

$$-3X^{2}+13X+10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3X^{2}+aX+bX+10. To find a and b, set up a system to be solved.

$$a+b=13 ab=-3\times 10=-30$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -30.

$$-1,30 -2,15 -3,10 -5,6$$

Calculate the sum for each pair.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

The solution is the pair that gives sum 13.

$$a=15 b=-2$$

Rewrite -3X^{2}+13X+10 as \left(-3X^{2}+15X\right)+\left(-2X+10\right).

$$\left(-3X^{2}+15X\right)+\left(-2X+10\right)$$

Factor out 3X in the first and 2 in the second group.

$$3X\left(-X+5\right)+2\left(-X+5\right)$$

Factor out common term -X+5 by using distributive property.

$$\left(-X+5\right)\left(3X+2\right)$$

To find equation solutions, solve -X+5=0 and 3X+2=0.

$$X=5 X=-\frac{2}{3}$$

Steps Using the Quadratic Formula

Variable X cannot be equal to any of the values -\frac{7}{4},\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2X-1\right)\left(4X+7\right), the least common multiple of 2X-1,4X+7.

$$\left(4X+7\right)\left(X+3\right)-\left(2X-1\right)\left(5X-1\right)=0$$

Use the distributive property to multiply 4X+7 by X+3 and combine like terms.

$$4X^{2}+19X+21-\left(2X-1\right)\left(5X-1\right)=0$$

Use the distributive property to multiply 2X-1 by 5X-1 and combine like terms.

$$4X^{2}+19X+21-\left(10X^{2}-7X+1\right)=0$$

To find the opposite of 10X^{2}-7X+1, find the opposite of each term.

$$4X^{2}+19X+21-10X^{2}+7X-1=0$$

Combine 4X^{2} and -10X^{2} to get -6X^{2}.

$$-6X^{2}+19X+21+7X-1=0$$

Combine 19X and 7X to get 26X.

$$-6X^{2}+26X+21-1=0$$

Subtract 1 from 21 to get 20.

$$-6X^{2}+26X+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 26 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$X=\frac{-26±\sqrt{26^{2}-4\left(-6\right)\times 20}}{2\left(-6\right)}$$

Square 26.

$$X=\frac{-26±\sqrt{676-4\left(-6\right)\times 20}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$X=\frac{-26±\sqrt{676+24\times 20}}{2\left(-6\right)}$$

Multiply 24 times 20.

$$X=\frac{-26±\sqrt{676+480}}{2\left(-6\right)}$$

Add 676 to 480.

$$X=\frac{-26±\sqrt{1156}}{2\left(-6\right)}$$

Take the square root of 1156.

$$X=\frac{-26±34}{2\left(-6\right)}$$

Multiply 2 times -6.

$$X=\frac{-26±34}{-12}$$

Now solve the equation X=\frac{-26±34}{-12} when ± is plus. Add -26 to 34.

$$X=\frac{8}{-12}$$

Reduce the fraction \frac{8}{-12} to lowest terms by extracting and canceling out 4.

$$X=-\frac{2}{3}$$

Now solve the equation X=\frac{-26±34}{-12} when ± is minus. Subtract 34 from -26.

$$X=-\frac{60}{-12}$$

Divide -60 by -12.

$$X=5$$

The equation is now solved.

$$X=-\frac{2}{3} X=5$$

Steps for Completing the Square

Variable X cannot be equal to any of the values -\frac{7}{4},\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2X-1\right)\left(4X+7\right), the least common multiple of 2X-1,4X+7.

$$\left(4X+7\right)\left(X+3\right)-\left(2X-1\right)\left(5X-1\right)=0$$

Use the distributive property to multiply 4X+7 by X+3 and combine like terms.

$$4X^{2}+19X+21-\left(2X-1\right)\left(5X-1\right)=0$$

Use the distributive property to multiply 2X-1 by 5X-1 and combine like terms.

$$4X^{2}+19X+21-\left(10X^{2}-7X+1\right)=0$$

To find the opposite of 10X^{2}-7X+1, find the opposite of each term.

$$4X^{2}+19X+21-10X^{2}+7X-1=0$$

Combine 4X^{2} and -10X^{2} to get -6X^{2}.

$$-6X^{2}+19X+21+7X-1=0$$

Combine 19X and 7X to get 26X.

$$-6X^{2}+26X+21-1=0$$

Subtract 1 from 21 to get 20.

$$-6X^{2}+26X+20=0$$

Subtract 20 from both sides. Anything subtracted from zero gives its negation.

$$-6X^{2}+26X=-20$$

Divide both sides by -6.

$$\frac{-6X^{2}+26X}{-6}=-\frac{20}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$X^{2}+\frac{26}{-6}X=-\frac{20}{-6}$$

Reduce the fraction \frac{26}{-6} to lowest terms by extracting and canceling out 2.

$$X^{2}-\frac{13}{3}X=-\frac{20}{-6}$$

Reduce the fraction \frac{-20}{-6} to lowest terms by extracting and canceling out 2.

$$X^{2}-\frac{13}{3}X=\frac{10}{3}$$

Divide -\frac{13}{3}, the coefficient of the x term, by 2 to get -\frac{13}{6}. Then add the square of -\frac{13}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$X^{2}-\frac{13}{3}X+\left(-\frac{13}{6}\right)^{2}=\frac{10}{3}+\left(-\frac{13}{6}\right)^{2}$$

Square -\frac{13}{6} by squaring both the numerator and the denominator of the fraction.

$$X^{2}-\frac{13}{3}X+\frac{169}{36}=\frac{10}{3}+\frac{169}{36}$$

Add \frac{10}{3} to \frac{169}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$X^{2}-\frac{13}{3}X+\frac{169}{36}=\frac{289}{36}$$

Factor X^{2}-\frac{13}{3}X+\frac{169}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(X-\frac{13}{6}\right)^{2}=\frac{289}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(X-\frac{13}{6}\right)^{2}}=\sqrt{\frac{289}{36}}$$

Simplify.

$$X-\frac{13}{6}=\frac{17}{6} X-\frac{13}{6}=-\frac{17}{6}$$

Add \frac{13}{6} to both sides of the equation.

$$X=5 X=-\frac{2}{3}$$