Solve 125-25a=-9a^2+54a+45

$$125-25a = -9 { a }^{ 2 } +54a+45$$

Solve for a

$$a = \frac{\sqrt{3361} + 79}{18} \approx 7.609674009$$

$$a = \frac{79 - \sqrt{3361}}{18} \approx 1.168103769$$

Steps Using the Quadratic Formula

Add 9a^{2} to both sides.

$$125-25a+9a^{2}=54a+45$$

Subtract 54a from both sides.

$$125-25a+9a^{2}-54a=45$$

Combine -25a and -54a to get -79a.

$$125-79a+9a^{2}=45$$

Subtract 45 from both sides.

$$125-79a+9a^{2}-45=0$$

Subtract 45 from 125 to get 80.

$$80-79a+9a^{2}=0$$

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.

$$9a^{2}-79a+80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -79 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-79\right)±\sqrt{\left(-79\right)^{2}-4\times 9\times 80}}{2\times 9}$$

Square -79.

$$a=\frac{-\left(-79\right)±\sqrt{6241-4\times 9\times 80}}{2\times 9}$$

Multiply -4 times 9.

$$a=\frac{-\left(-79\right)±\sqrt{6241-36\times 80}}{2\times 9}$$

Multiply -36 times 80.

$$a=\frac{-\left(-79\right)±\sqrt{6241-2880}}{2\times 9}$$

Add 6241 to -2880.

$$a=\frac{-\left(-79\right)±\sqrt{3361}}{2\times 9}$$

The opposite of -79 is 79.

$$a=\frac{79±\sqrt{3361}}{2\times 9}$$

Multiply 2 times 9.

$$a=\frac{79±\sqrt{3361}}{18}$$

Now solve the equation a=\frac{79±\sqrt{3361}}{18} when ± is plus. Add 79 to \sqrt{3361}.

$$a=\frac{\sqrt{3361}+79}{18}$$

Now solve the equation a=\frac{79±\sqrt{3361}}{18} when ± is minus. Subtract \sqrt{3361} from 79.

$$a=\frac{79-\sqrt{3361}}{18}$$

The equation is now solved.

$$a=\frac{\sqrt{3361}+79}{18} a=\frac{79-\sqrt{3361}}{18}$$

Steps for Completing the Square

Add 9a^{2} to both sides.

$$125-25a+9a^{2}=54a+45$$

Subtract 54a from both sides.

$$125-25a+9a^{2}-54a=45$$

Combine -25a and -54a to get -79a.

$$125-79a+9a^{2}=45$$

Subtract 125 from both sides.

$$-79a+9a^{2}=45-125$$

Subtract 125 from 45 to get -80.

$$-79a+9a^{2}=-80$$

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.

$$9a^{2}-79a=-80$$

Divide both sides by 9.

$$\frac{9a^{2}-79a}{9}=-\frac{80}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$a^{2}-\frac{79}{9}a=-\frac{80}{9}$$

Divide -\frac{79}{9}, the coefficient of the x term, by 2 to get -\frac{79}{18}. Then add the square of -\frac{79}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{79}{9}a+\left(-\frac{79}{18}\right)^{2}=-\frac{80}{9}+\left(-\frac{79}{18}\right)^{2}$$

Square -\frac{79}{18} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{79}{9}a+\frac{6241}{324}=-\frac{80}{9}+\frac{6241}{324}$$

Add -\frac{80}{9} to \frac{6241}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{79}{9}a+\frac{6241}{324}=\frac{3361}{324}$$

Factor a^{2}-\frac{79}{9}a+\frac{6241}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{79}{18}\right)^{2}=\frac{3361}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{79}{18}\right)^{2}}=\sqrt{\frac{3361}{324}}$$

Simplify.

$$a-\frac{79}{18}=\frac{\sqrt{3361}}{18} a-\frac{79}{18}=-\frac{\sqrt{3361}}{18}$$

Add \frac{79}{18} to both sides of the equation.

$$a=\frac{\sqrt{3361}+79}{18} a=\frac{79-\sqrt{3361}}{18}$$