Solve {l}{text{socumons}}{2x/2x+5=3/4-7/4x+10}

$$\left. \begin{array} { l } { \text { socumons } } \\ { \frac { 2 x } { 2 x + 5 } = \frac { 3 } { 4 } - \frac { 7 } { 4 x + 10 } } \end{array} \right.$$

Solve for x

$$x=\frac{-37+\sqrt{215}i}{16}\approx -2.3125+0.916429894i$$

$$x=\frac{-\sqrt{215}i-37}{16}\approx -2.3125-0.916429894i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -\frac{5}{2} since division by zero is not defined. Multiply both sides of the equation by 4\left(2x+5\right), the least common multiple of 4,4x+10.

$$2x\times 4\left(2x+5\right)+4\left(2x+5\right)\times 5=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Multiply 2 and 4 to get 8.

$$8x\left(2x+5\right)+4\left(2x+5\right)\times 5=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Use the distributive property to multiply 8x by 2x+5.

$$16x^{2}+40x+4\left(2x+5\right)\times 5=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Multiply 4 and 5 to get 20.

$$16x^{2}+40x+20\left(2x+5\right)=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Use the distributive property to multiply 20 by 2x+5.

$$16x^{2}+40x+40x+100=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Combine 40x and 40x to get 80x.

$$16x^{2}+80x+100=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Multiply 4 and \frac{3}{4} to get 3.

$$16x^{2}+80x+100=3\left(2x+5\right)-2\times 7$$

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

$$16x^{2}+80x+100=6x+15-2\times 7$$

Multiply -2 and 7 to get -14.

$$16x^{2}+80x+100=6x+15-14$$

Subtract 14 from 15 to get 1.

$$16x^{2}+80x+100=6x+1$$

Subtract 6x from both sides.

$$16x^{2}+80x+100-6x=1$$

Combine 80x and -6x to get 74x.

$$16x^{2}+74x+100=1$$

Subtract 1 from both sides.

$$16x^{2}+74x+100-1=0$$

Subtract 1 from 100 to get 99.

$$16x^{2}+74x+99=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 74 for b, and 99 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-74±\sqrt{74^{2}-4\times 16\times 99}}{2\times 16}$$

Square 74.

$$x=\frac{-74±\sqrt{5476-4\times 16\times 99}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-74±\sqrt{5476-64\times 99}}{2\times 16}$$

Multiply -64 times 99.

$$x=\frac{-74±\sqrt{5476-6336}}{2\times 16}$$

Add 5476 to -6336.

$$x=\frac{-74±\sqrt{-860}}{2\times 16}$$

Take the square root of -860.

$$x=\frac{-74±2\sqrt{215}i}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-74±2\sqrt{215}i}{32}$$

Now solve the equation x=\frac{-74±2\sqrt{215}i}{32} when ± is plus. Add -74 to 2i\sqrt{215}.

$$x=\frac{-74+2\sqrt{215}i}{32}$$

Divide -74+2i\sqrt{215} by 32.

$$x=\frac{-37+\sqrt{215}i}{16}$$

Now solve the equation x=\frac{-74±2\sqrt{215}i}{32} when ± is minus. Subtract 2i\sqrt{215} from -74.

$$x=\frac{-2\sqrt{215}i-74}{32}$$

Divide -74-2i\sqrt{215} by 32.

$$x=\frac{-\sqrt{215}i-37}{16}$$

The equation is now solved.

$$x=\frac{-37+\sqrt{215}i}{16} x=\frac{-\sqrt{215}i-37}{16}$$

Steps for Completing the Square

Variable x cannot be equal to -\frac{5}{2} since division by zero is not defined. Multiply both sides of the equation by 4\left(2x+5\right), the least common multiple of 4,4x+10.

$$2x\times 4\left(2x+5\right)+4\left(2x+5\right)\times 5=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Multiply 2 and 4 to get 8.

$$8x\left(2x+5\right)+4\left(2x+5\right)\times 5=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Use the distributive property to multiply 8x by 2x+5.

$$16x^{2}+40x+4\left(2x+5\right)\times 5=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Multiply 4 and 5 to get 20.

$$16x^{2}+40x+20\left(2x+5\right)=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Use the distributive property to multiply 20 by 2x+5.

$$16x^{2}+40x+40x+100=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Combine 40x and 40x to get 80x.

$$16x^{2}+80x+100=4\left(2x+5\right)\times \frac{3}{4}-2\times 7$$

Multiply 4 and \frac{3}{4} to get 3.

$$16x^{2}+80x+100=3\left(2x+5\right)-2\times 7$$

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

$$16x^{2}+80x+100=6x+15-2\times 7$$

Multiply -2 and 7 to get -14.

$$16x^{2}+80x+100=6x+15-14$$

Subtract 14 from 15 to get 1.

$$16x^{2}+80x+100=6x+1$$

Subtract 6x from both sides.

$$16x^{2}+80x+100-6x=1$$

Combine 80x and -6x to get 74x.

$$16x^{2}+74x+100=1$$

Subtract 100 from both sides.

$$16x^{2}+74x=1-100$$

Subtract 100 from 1 to get -99.

$$16x^{2}+74x=-99$$

Divide both sides by 16.

$$\frac{16x^{2}+74x}{16}=-\frac{99}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\frac{74}{16}x=-\frac{99}{16}$$

Reduce the fraction \frac{74}{16} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{37}{8}x=-\frac{99}{16}$$

Divide \frac{37}{8}, the coefficient of the x term, by 2 to get \frac{37}{16}. Then add the square of \frac{37}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{37}{8}x+\left(\frac{37}{16}\right)^{2}=-\frac{99}{16}+\left(\frac{37}{16}\right)^{2}$$

Square \frac{37}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{37}{8}x+\frac{1369}{256}=-\frac{99}{16}+\frac{1369}{256}$$

Add -\frac{99}{16} to \frac{1369}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{37}{8}x+\frac{1369}{256}=-\frac{215}{256}$$

Factor x^{2}+\frac{37}{8}x+\frac{1369}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{37}{16}\right)^{2}=-\frac{215}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{37}{16}\right)^{2}}=\sqrt{-\frac{215}{256}}$$

Simplify.

$$x+\frac{37}{16}=\frac{\sqrt{215}i}{16} x+\frac{37}{16}=-\frac{\sqrt{215}i}{16}$$

Subtract \frac{37}{16} from both sides of the equation.

$$x=\frac{-37+\sqrt{215}i}{16} x=\frac{-\sqrt{215}i-37}{16}$$