Solve a/a-6=-21/a-10

$$\frac { a } { a - 6 } = \frac { - 21 } { a - 10 }$$

Solve for a

$$a=-18$$

$$a=7$$

Steps Using the Quadratic Formula

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

$$\left(a-10\right)a=\left(a-6\right)\left(-21\right)$$

Use the distributive property to multiply a-10 by a.

$$a^{2}-10a=\left(a-6\right)\left(-21\right)$$

Use the distributive property to multiply a-6 by -21.

$$a^{2}-10a=-21a+126$$

Add 21a to both sides.

$$a^{2}-10a+21a=126$$

Combine -10a and 21a to get 11a.

$$a^{2}+11a=126$$

Subtract 126 from both sides.

$$a^{2}+11a-126=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 11 for b, and -126 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-11±\sqrt{11^{2}-4\left(-126\right)}}{2}$$

Square 11.

$$a=\frac{-11±\sqrt{121-4\left(-126\right)}}{2}$$

Multiply -4 times -126.

$$a=\frac{-11±\sqrt{121+504}}{2}$$

Add 121 to 504.

$$a=\frac{-11±\sqrt{625}}{2}$$

Take the square root of 625.

$$a=\frac{-11±25}{2}$$

Now solve the equation a=\frac{-11±25}{2} when ± is plus. Add -11 to 25.

$$a=\frac{14}{2}$$

Divide 14 by 2.

$$a=7$$

Now solve the equation a=\frac{-11±25}{2} when ± is minus. Subtract 25 from -11.

$$a=-\frac{36}{2}$$

Divide -36 by 2.

$$a=-18$$

The equation is now solved.

$$a=7 a=-18$$

Steps for Completing the Square

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

$$\left(a-10\right)a=\left(a-6\right)\left(-21\right)$$

Use the distributive property to multiply a-10 by a.

$$a^{2}-10a=\left(a-6\right)\left(-21\right)$$

Use the distributive property to multiply a-6 by -21.

$$a^{2}-10a=-21a+126$$

Add 21a to both sides.

$$a^{2}-10a+21a=126$$

Combine -10a and 21a to get 11a.

$$a^{2}+11a=126$$

Divide 11, the coefficient of the x term, by 2 to get \frac{11}{2}. Then add the square of \frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+11a+\left(\frac{11}{2}\right)^{2}=126+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+11a+\frac{121}{4}=126+\frac{121}{4}$$

Add 126 to \frac{121}{4}.

$$a^{2}+11a+\frac{121}{4}=\frac{625}{4}$$

Factor a^{2}+11a+\frac{121}{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}.

$$\left(a+\frac{11}{2}\right)^{2}=\frac{625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{11}{2}\right)^{2}}=\sqrt{\frac{625}{4}}$$

Simplify.

$$a+\frac{11}{2}=\frac{25}{2} a+\frac{11}{2}=-\frac{25}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$a=7 a=-18$$