Solve 6/2x+4=x-5/10

$$\frac { 6 } { 2 x + 4 } = \frac { x - 5 } { 10 }$$

Solve for x

$$x=-5$$

$$x=8$$

Steps Using the Quadratic Formula

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

$$5\times 6=\left(x+2\right)\left(x-5\right)$$

Multiply 5 and 6 to get 30.

$$30=\left(x+2\right)\left(x-5\right)$$

Use the distributive property to multiply x+2 by x-5 and combine like terms.

$$30=x^{2}-3x-10$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-3x-10=30$$

Subtract 30 from both sides.

$$x^{2}-3x-10-30=0$$

Subtract 30 from -10 to get -40.

$$x^{2}-3x-40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-40\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-40\right)}}{2}$$

Multiply -4 times -40.

$$x=\frac{-\left(-3\right)±\sqrt{9+160}}{2}$$

Add 9 to 160.

$$x=\frac{-\left(-3\right)±\sqrt{169}}{2}$$

Take the square root of 169.

$$x=\frac{-\left(-3\right)±13}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±13}{2}$$

Now solve the equation x=\frac{3±13}{2} when ± is plus. Add 3 to 13.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{3±13}{2} when ± is minus. Subtract 13 from 3.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

The equation is now solved.

$$x=8 x=-5$$

Steps for Completing the Square

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

$$5\times 6=\left(x+2\right)\left(x-5\right)$$

Multiply 5 and 6 to get 30.

$$30=\left(x+2\right)\left(x-5\right)$$

Use the distributive property to multiply x+2 by x-5 and combine like terms.

$$30=x^{2}-3x-10$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-3x-10=30$$

Add 10 to both sides.

$$x^{2}-3x=30+10$$

Add 30 and 10 to get 40.

$$x^{2}-3x=40$$

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

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=40+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=40+\frac{9}{4}$$

Add 40 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{169}{4}$$

Factor x^{2}-3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{13}{2} x-\frac{3}{2}=-\frac{13}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=8 x=-5$$