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

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

Solve for x

$$x=-5$$

$$x=8$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-5\right), the least common multiple of 6,x-5.

$$\left(x-5\right)\left(2x+4\right)=6\times 10$$

Use the distributive property to multiply x-5 by 2x+4 and combine like terms.

$$2x^{2}-6x-20=6\times 10$$

Multiply 6 and 10 to get 60.

$$2x^{2}-6x-20=60$$

Subtract 60 from both sides.

$$2x^{2}-6x-20-60=0$$

Subtract 60 from -20 to get -80.

$$2x^{2}-6x-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -6 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\left(-80\right)}}{2\times 2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 2\left(-80\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-6\right)±\sqrt{36-8\left(-80\right)}}{2\times 2}$$

Multiply -8 times -80.

$$x=\frac{-\left(-6\right)±\sqrt{36+640}}{2\times 2}$$

Add 36 to 640.

$$x=\frac{-\left(-6\right)±\sqrt{676}}{2\times 2}$$

Take the square root of 676.

$$x=\frac{-\left(-6\right)±26}{2\times 2}$$

The opposite of -6 is 6.

$$x=\frac{6±26}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{6±26}{4}$$

Now solve the equation x=\frac{6±26}{4} when ± is plus. Add 6 to 26.

$$x=\frac{32}{4}$$

Divide 32 by 4.

$$x=8$$

Now solve the equation x=\frac{6±26}{4} when ± is minus. Subtract 26 from 6.

$$x=-\frac{20}{4}$$

Divide -20 by 4.

$$x=-5$$

The equation is now solved.

$$x=8 x=-5$$

Steps for Completing the Square

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by 6\left(x-5\right), the least common multiple of 6,x-5.

$$\left(x-5\right)\left(2x+4\right)=6\times 10$$

Use the distributive property to multiply x-5 by 2x+4 and combine like terms.

$$2x^{2}-6x-20=6\times 10$$

Multiply 6 and 10 to get 60.

$$2x^{2}-6x-20=60$$

Add 20 to both sides.

$$2x^{2}-6x=60+20$$

Add 60 and 20 to get 80.

$$2x^{2}-6x=80$$

Divide both sides by 2.

$$\frac{2x^{2}-6x}{2}=\frac{80}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{6}{2}\right)x=\frac{80}{2}$$

Divide -6 by 2.

$$x^{2}-3x=\frac{80}{2}$$

Divide 80 by 2.

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

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.

$$x=8 x=-5$$