Solve 7/x+2+10/2x-5=3

$$\frac{ 7 }{ x+2 } + \frac{ 10 }{ 2x-5 } =3$$

Solve for x

$$x=5$$

$$x=-\frac{1}{2}=-0.5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,\frac{5}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-5\right)\left(x+2\right), the least common multiple of x+2,2x-5.

$$\left(2x-5\right)\times 7+\left(x+2\right)\times 10=3\left(2x-5\right)\left(x+2\right)$$

Use the distributive property to multiply 2x-5 by 7.

$$14x-35+\left(x+2\right)\times 10=3\left(2x-5\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 10.

$$14x-35+10x+20=3\left(2x-5\right)\left(x+2\right)$$

Combine 14x and 10x to get 24x.

$$24x-35+20=3\left(2x-5\right)\left(x+2\right)$$

Add -35 and 20 to get -15.

$$24x-15=3\left(2x-5\right)\left(x+2\right)$$

Use the distributive property to multiply 3 by 2x-5.

$$24x-15=\left(6x-15\right)\left(x+2\right)$$

Use the distributive property to multiply 6x-15 by x+2 and combine like terms.

$$24x-15=6x^{2}-3x-30$$

Subtract 6x^{2} from both sides.

$$24x-15-6x^{2}=-3x-30$$

Add 3x to both sides.

$$24x-15-6x^{2}+3x=-30$$

Combine 24x and 3x to get 27x.

$$27x-15-6x^{2}=-30$$

Add 30 to both sides.

$$27x-15-6x^{2}+30=0$$

Add -15 and 30 to get 15.

$$27x+15-6x^{2}=0$$

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.

$$-6x^{2}+27x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 27 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27±\sqrt{27^{2}-4\left(-6\right)\times 15}}{2\left(-6\right)}$$

Square 27.

$$x=\frac{-27±\sqrt{729-4\left(-6\right)\times 15}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-27±\sqrt{729+24\times 15}}{2\left(-6\right)}$$

Multiply 24 times 15.

$$x=\frac{-27±\sqrt{729+360}}{2\left(-6\right)}$$

Add 729 to 360.

$$x=\frac{-27±\sqrt{1089}}{2\left(-6\right)}$$

Take the square root of 1089.

$$x=\frac{-27±33}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-27±33}{-12}$$

Now solve the equation x=\frac{-27±33}{-12} when ± is plus. Add -27 to 33.

$$x=\frac{6}{-12}$$

Reduce the fraction \frac{6}{-12} to lowest terms by extracting and canceling out 6.

$$x=-\frac{1}{2}$$

Now solve the equation x=\frac{-27±33}{-12} when ± is minus. Subtract 33 from -27.

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

Divide -60 by -12.

$$x=5$$

The equation is now solved.

$$x=-\frac{1}{2} x=5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,\frac{5}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-5\right)\left(x+2\right), the least common multiple of x+2,2x-5.

$$\left(2x-5\right)\times 7+\left(x+2\right)\times 10=3\left(2x-5\right)\left(x+2\right)$$

Use the distributive property to multiply 2x-5 by 7.

$$14x-35+\left(x+2\right)\times 10=3\left(2x-5\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 10.

$$14x-35+10x+20=3\left(2x-5\right)\left(x+2\right)$$

Combine 14x and 10x to get 24x.

$$24x-35+20=3\left(2x-5\right)\left(x+2\right)$$

Add -35 and 20 to get -15.

$$24x-15=3\left(2x-5\right)\left(x+2\right)$$

Use the distributive property to multiply 3 by 2x-5.

$$24x-15=\left(6x-15\right)\left(x+2\right)$$

Use the distributive property to multiply 6x-15 by x+2 and combine like terms.

$$24x-15=6x^{2}-3x-30$$

Subtract 6x^{2} from both sides.

$$24x-15-6x^{2}=-3x-30$$

Add 3x to both sides.

$$24x-15-6x^{2}+3x=-30$$

Combine 24x and 3x to get 27x.

$$27x-15-6x^{2}=-30$$

Add 15 to both sides.

$$27x-6x^{2}=-30+15$$

Add -30 and 15 to get -15.

$$27x-6x^{2}=-15$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-6x^{2}+27x=-15$$

Divide both sides by -6.

$$\frac{-6x^{2}+27x}{-6}=-\frac{15}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{27}{-6}x=-\frac{15}{-6}$$

Reduce the fraction \frac{27}{-6} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{9}{2}x=-\frac{15}{-6}$$

Reduce the fraction \frac{-15}{-6} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{9}{2}x=\frac{5}{2}$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=\frac{5}{2}+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{5}{2}+\frac{81}{16}$$

Add \frac{5}{2} to \frac{81}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{121}{16}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{121}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{121}{16}}$$

Simplify.

$$x-\frac{9}{4}=\frac{11}{4} x-\frac{9}{4}=-\frac{11}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$x=5 x=-\frac{1}{2}$$