Solve 7+x/10=2/x+6

$$\frac{ 7+x }{ 10 } = \frac{ 2 }{ x+6 }$$

Solve for x

$$x=-11$$

$$x=-2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -6 since division by zero is not defined. Multiply both sides of the equation by 10\left(x+6\right), the least common multiple of 10,x+6.

$$\left(x+6\right)\left(7+x\right)=10\times 2$$

Use the distributive property to multiply x+6 by 7+x and combine like terms.

$$13x+x^{2}+42=10\times 2$$

Multiply 10 and 2 to get 20.

$$13x+x^{2}+42=20$$

Subtract 20 from both sides.

$$13x+x^{2}+42-20=0$$

Subtract 20 from 42 to get 22.

$$13x+x^{2}+22=0$$

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.

$$x^{2}+13x+22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 13 for b, and 22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 22}}{2}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 22}}{2}$$

Multiply -4 times 22.

$$x=\frac{-13±\sqrt{169-88}}{2}$$

Add 169 to -88.

$$x=\frac{-13±\sqrt{81}}{2}$$

Take the square root of 81.

$$x=\frac{-13±9}{2}$$

Now solve the equation x=\frac{-13±9}{2} when ± is plus. Add -13 to 9.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

Now solve the equation x=\frac{-13±9}{2} when ± is minus. Subtract 9 from -13.

$$x=-\frac{22}{2}$$

Divide -22 by 2.

$$x=-11$$

The equation is now solved.

$$x=-2 x=-11$$

Steps for Completing the Square

Variable x cannot be equal to -6 since division by zero is not defined. Multiply both sides of the equation by 10\left(x+6\right), the least common multiple of 10,x+6.

$$\left(x+6\right)\left(7+x\right)=10\times 2$$

Use the distributive property to multiply x+6 by 7+x and combine like terms.

$$13x+x^{2}+42=10\times 2$$

Multiply 10 and 2 to get 20.

$$13x+x^{2}+42=20$$

Subtract 42 from both sides.

$$13x+x^{2}=20-42$$

Subtract 42 from 20 to get -22.

$$13x+x^{2}=-22$$

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.

$$x^{2}+13x=-22$$

Divide 13, the coefficient of the x term, by 2 to get \frac{13}{2}. Then add the square of \frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+13x+\left(\frac{13}{2}\right)^{2}=-22+\left(\frac{13}{2}\right)^{2}$$

Square \frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+13x+\frac{169}{4}=-22+\frac{169}{4}$$

Add -22 to \frac{169}{4}.

$$x^{2}+13x+\frac{169}{4}=\frac{81}{4}$$

Factor x^{2}+13x+\frac{169}{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}.

$$\left(x+\frac{13}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x+\frac{13}{2}=\frac{9}{2} x+\frac{13}{2}=-\frac{9}{2}$$

Subtract \frac{13}{2} from both sides of the equation.

$$x=-2 x=-11$$