Solve x+3/2x-1-5x-1/4x+7=0

$$\frac { x + 3 } { 2 x - 1 } - \frac { 5 x - 1 } { 4 x + 7 } = 0$$

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.

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