Solve frac{2a_{2}}{5-a_{2}}=frac{21-a_{2}}{2a_{2}}

$$\frac { 2 a _ { 2 } } { 5 - a _ { 2 } } = \frac { 21 - a _ { 2 } } { 2 a _ { 2 } }$$

Solve for a_2

$$a_{2} = -\frac{35}{3} = -11\frac{2}{3} \approx -11.666666667$$

$$a_{2}=3$$

Steps Using the Quadratic Formula

Variable a_{2} cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by 2a_{2}\left(a_{2}-5\right), the least common multiple of 5-a_{2},2a_{2}.

$$-2a_{2}\times 2a_{2}=\left(a_{2}-5\right)\left(21-a_{2}\right)$$

Multiply -2 and 2 to get -4.

$$-4a_{2}a_{2}=\left(a_{2}-5\right)\left(21-a_{2}\right)$$

Multiply a_{2} and a_{2} to get a_{2}^{2}.

$$-4a_{2}^{2}=\left(a_{2}-5\right)\left(21-a_{2}\right)$$

Use the distributive property to multiply a_{2}-5 by 21-a_{2} and combine like terms.

$$-4a_{2}^{2}=26a_{2}-a_{2}^{2}-105$$

Subtract 26a_{2} from both sides.

$$-4a_{2}^{2}-26a_{2}=-a_{2}^{2}-105$$

Add a_{2}^{2} to both sides.

$$-4a_{2}^{2}-26a_{2}+a_{2}^{2}=-105$$

Combine -4a_{2}^{2} and a_{2}^{2} to get -3a_{2}^{2}.

$$-3a_{2}^{2}-26a_{2}=-105$$

Add 105 to both sides.

$$-3a_{2}^{2}-26a_{2}+105=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -26 for b, and 105 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a_{2}=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\left(-3\right)\times 105}}{2\left(-3\right)}$$

Square -26.

$$a_{2}=\frac{-\left(-26\right)±\sqrt{676-4\left(-3\right)\times 105}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$a_{2}=\frac{-\left(-26\right)±\sqrt{676+12\times 105}}{2\left(-3\right)}$$

Multiply 12 times 105.

$$a_{2}=\frac{-\left(-26\right)±\sqrt{676+1260}}{2\left(-3\right)}$$

Add 676 to 1260.

$$a_{2}=\frac{-\left(-26\right)±\sqrt{1936}}{2\left(-3\right)}$$

Take the square root of 1936.

$$a_{2}=\frac{-\left(-26\right)±44}{2\left(-3\right)}$$

The opposite of -26 is 26.

$$a_{2}=\frac{26±44}{2\left(-3\right)}$$

Multiply 2 times -3.

$$a_{2}=\frac{26±44}{-6}$$

Now solve the equation a_{2}=\frac{26±44}{-6} when ± is plus. Add 26 to 44.

$$a_{2}=\frac{70}{-6}$$

Reduce the fraction \frac{70}{-6} to lowest terms by extracting and canceling out 2.

$$a_{2}=-\frac{35}{3}$$

Now solve the equation a_{2}=\frac{26±44}{-6} when ± is minus. Subtract 44 from 26.

$$a_{2}=-\frac{18}{-6}$$

Divide -18 by -6.

$$a_{2}=3$$

The equation is now solved.

$$a_{2}=-\frac{35}{3} a_{2}=3$$

Steps for Completing the Square

Variable a_{2} cannot be equal to any of the values 0,5 since division by zero is not defined. Multiply both sides of the equation by 2a_{2}\left(a_{2}-5\right), the least common multiple of 5-a_{2},2a_{2}.

$$-2a_{2}\times 2a_{2}=\left(a_{2}-5\right)\left(21-a_{2}\right)$$

Multiply -2 and 2 to get -4.

$$-4a_{2}a_{2}=\left(a_{2}-5\right)\left(21-a_{2}\right)$$

Multiply a_{2} and a_{2} to get a_{2}^{2}.

$$-4a_{2}^{2}=\left(a_{2}-5\right)\left(21-a_{2}\right)$$

Use the distributive property to multiply a_{2}-5 by 21-a_{2} and combine like terms.

$$-4a_{2}^{2}=26a_{2}-a_{2}^{2}-105$$

Subtract 26a_{2} from both sides.

$$-4a_{2}^{2}-26a_{2}=-a_{2}^{2}-105$$

Add a_{2}^{2} to both sides.

$$-4a_{2}^{2}-26a_{2}+a_{2}^{2}=-105$$

Combine -4a_{2}^{2} and a_{2}^{2} to get -3a_{2}^{2}.

$$-3a_{2}^{2}-26a_{2}=-105$$

Divide both sides by -3.

$$\frac{-3a_{2}^{2}-26a_{2}}{-3}=-\frac{105}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$a_{2}^{2}+\left(-\frac{26}{-3}\right)a_{2}=-\frac{105}{-3}$$

Divide -26 by -3.

$$a_{2}^{2}+\frac{26}{3}a_{2}=-\frac{105}{-3}$$

Divide -105 by -3.

$$a_{2}^{2}+\frac{26}{3}a_{2}=35$$

Divide \frac{26}{3}, the coefficient of the x term, by 2 to get \frac{13}{3}. Then add the square of \frac{13}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a_{2}^{2}+\frac{26}{3}a_{2}+\left(\frac{13}{3}\right)^{2}=35+\left(\frac{13}{3}\right)^{2}$$

Square \frac{13}{3} by squaring both the numerator and the denominator of the fraction.

$$a_{2}^{2}+\frac{26}{3}a_{2}+\frac{169}{9}=35+\frac{169}{9}$$

Add 35 to \frac{169}{9}.

$$a_{2}^{2}+\frac{26}{3}a_{2}+\frac{169}{9}=\frac{484}{9}$$

Factor a_{2}^{2}+\frac{26}{3}a_{2}+\frac{169}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a_{2}+\frac{13}{3}\right)^{2}=\frac{484}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a_{2}+\frac{13}{3}\right)^{2}}=\sqrt{\frac{484}{9}}$$

Simplify.

$$a_{2}+\frac{13}{3}=\frac{22}{3} a_{2}+\frac{13}{3}=-\frac{22}{3}$$

Subtract \frac{13}{3} from both sides of the equation.

$$a_{2}=3 a_{2}=-\frac{35}{3}$$