Solve (a+5)(-5a+3)=-63^2

$$( a + 5 ) ( - 5 a + 3 ) = - 63 ^ { 2 }$$

Solve for a

$$a = \frac{7 \sqrt{409} - 11}{5} \approx 26.113247783$$

$$a=\frac{-7\sqrt{409}-11}{5}\approx -30.513247783$$

Steps Using the Quadratic Formula

Use the distributive property to multiply a+5 by -5a+3 and combine like terms.

$$-5a^{2}-22a+15=-63^{2}$$

Calculate 63 to the power of 2 and get 3969.

$$-5a^{2}-22a+15=-3969$$

Add 3969 to both sides.

$$-5a^{2}-22a+15+3969=0$$

Add 15 and 3969 to get 3984.

$$-5a^{2}-22a+3984=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -22 for b, and 3984 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-5\right)\times 3984}}{2\left(-5\right)}$$

Square -22.

$$a=\frac{-\left(-22\right)±\sqrt{484-4\left(-5\right)\times 3984}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$a=\frac{-\left(-22\right)±\sqrt{484+20\times 3984}}{2\left(-5\right)}$$

Multiply 20 times 3984.

$$a=\frac{-\left(-22\right)±\sqrt{484+79680}}{2\left(-5\right)}$$

Add 484 to 79680.

$$a=\frac{-\left(-22\right)±\sqrt{80164}}{2\left(-5\right)}$$

Take the square root of 80164.

$$a=\frac{-\left(-22\right)±14\sqrt{409}}{2\left(-5\right)}$$

The opposite of -22 is 22.

$$a=\frac{22±14\sqrt{409}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$a=\frac{22±14\sqrt{409}}{-10}$$

Now solve the equation a=\frac{22±14\sqrt{409}}{-10} when ± is plus. Add 22 to 14\sqrt{409}.

$$a=\frac{14\sqrt{409}+22}{-10}$$

Divide 22+14\sqrt{409} by -10.

$$a=\frac{-7\sqrt{409}-11}{5}$$

Now solve the equation a=\frac{22±14\sqrt{409}}{-10} when ± is minus. Subtract 14\sqrt{409} from 22.

$$a=\frac{22-14\sqrt{409}}{-10}$$

Divide 22-14\sqrt{409} by -10.

$$a=\frac{7\sqrt{409}-11}{5}$$

The equation is now solved.

$$a=\frac{-7\sqrt{409}-11}{5} a=\frac{7\sqrt{409}-11}{5}$$

Steps for Completing the Square

Use the distributive property to multiply a+5 by -5a+3 and combine like terms.

$$-5a^{2}-22a+15=-63^{2}$$

Calculate 63 to the power of 2 and get 3969.

$$-5a^{2}-22a+15=-3969$$

Subtract 15 from both sides.

$$-5a^{2}-22a=-3969-15$$

Subtract 15 from -3969 to get -3984.

$$-5a^{2}-22a=-3984$$

Divide both sides by -5.

$$\frac{-5a^{2}-22a}{-5}=-\frac{3984}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$a^{2}+\left(-\frac{22}{-5}\right)a=-\frac{3984}{-5}$$

Divide -22 by -5.

$$a^{2}+\frac{22}{5}a=-\frac{3984}{-5}$$

Divide -3984 by -5.

$$a^{2}+\frac{22}{5}a=\frac{3984}{5}$$

Divide \frac{22}{5}, the coefficient of the x term, by 2 to get \frac{11}{5}. Then add the square of \frac{11}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{22}{5}a+\left(\frac{11}{5}\right)^{2}=\frac{3984}{5}+\left(\frac{11}{5}\right)^{2}$$

Square \frac{11}{5} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{22}{5}a+\frac{121}{25}=\frac{3984}{5}+\frac{121}{25}$$

Add \frac{3984}{5} to \frac{121}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{22}{5}a+\frac{121}{25}=\frac{20041}{25}$$

Factor a^{2}+\frac{22}{5}a+\frac{121}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{11}{5}\right)^{2}=\frac{20041}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{11}{5}\right)^{2}}=\sqrt{\frac{20041}{25}}$$

Simplify.

$$a+\frac{11}{5}=\frac{7\sqrt{409}}{5} a+\frac{11}{5}=-\frac{7\sqrt{409}}{5}$$

Subtract \frac{11}{5} from both sides of the equation.

$$a=\frac{7\sqrt{409}-11}{5} a=\frac{-7\sqrt{409}-11}{5}$$