Solve 25/16a^2+25/18a+25/15=9/16a^2+frac{225}{16}-frac{45a}{8}

$$\frac{ 25 }{ 16 } { a }^{ 2 } + \frac{ 25 }{ 18 } a+ \frac{ 25 }{ 15 } = \frac{ 9 }{ 16 } { a }^{ 2 } + \frac{ 225 }{ 16 } - \frac{ 45a }{ 8 }$$

Solve for a

$$a = \frac{\sqrt{512065} - 505}{144} \approx 1.462410913$$

$$a=\frac{-\sqrt{512065}-505}{144}\approx -8.476299802$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 720, the least common multiple of 16,18,15,8.

$$1125a^{2}+1000a+48\times 25=405a^{2}+10125-90\times 45a$$

Multiply 48 and 25 to get 1200.

$$1125a^{2}+1000a+1200=405a^{2}+10125-90\times 45a$$

Multiply -90 and 45 to get -4050.

$$1125a^{2}+1000a+1200=405a^{2}[phone]a$$

Subtract 405a^{2} from both sides.

$$1125a^{2}+1000a+1200-405a^{2}=10125-4050a$$

Combine 1125a^{2} and -405a^{2} to get 720a^{2}.

$$720a^{2}+1000a+1200=10125-4050a$$

Subtract 10125 from both sides.

$$720a^{2}+1000a+1200-10125=-4050a$$

Subtract 10125 from 1200 to get -8925.

$$720a^{2}+1000a-8925=-4050a$$

Add 4050a to both sides.

$$720a^{2}+1000a-8925+4050a=0$$

Combine 1000a and 4050a to get 5050a.

$$720a^{2}+5050a-8925=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 720 for a, 5050 for b, and -8925 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-5050±\sqrt{5050^{2}-4\times 720\left(-8925\right)}}{2\times 720}$$

Square 5050.

$$a=\frac{-5050±\sqrt{25502500-4\times 720\left(-8925\right)}}{2\times 720}$$

Multiply -4 times 720.

$$a=\frac{-5050±\sqrt{25502500-2880\left(-8925\right)}}{2\times 720}$$

Multiply -2880 times -8925.

$$a=\frac{-5050±\sqrt{25502500+25704000}}{2\times 720}$$

Add 25502500 to 25704000.

$$a=\frac{-5050±\sqrt{51206500}}{2\times 720}$$

Take the square root of 51206500.

$$a=\frac{-5050±10\sqrt{512065}}{2\times 720}$$

Multiply 2 times 720.

$$a=\frac{-5050±10\sqrt{512065}}{1440}$$

Now solve the equation a=\frac{-5050±10\sqrt{512065}}{1440} when ± is plus. Add -5050 to 10\sqrt{512065}.

$$a=\frac{10\sqrt{512065}-5050}{1440}$$

Divide -5050+10\sqrt{512065} by 1440.

$$a=\frac{\sqrt{512065}-505}{144}$$

Now solve the equation a=\frac{-5050±10\sqrt{512065}}{1440} when ± is minus. Subtract 10\sqrt{512065} from -5050.

$$a=\frac{-10\sqrt{512065}-5050}{1440}$$

Divide -5050-10\sqrt{512065} by 1440.

$$a=\frac{-\sqrt{512065}-505}{144}$$

The equation is now solved.

$$a=\frac{\sqrt{512065}-505}{144} a=\frac{-\sqrt{512065}-505}{144}$$

Steps for Completing the Square

Multiply both sides of the equation by 720, the least common multiple of 16,18,15,8.

$$1125a^{2}+1000a+48\times 25=405a^{2}+10125-90\times 45a$$

Multiply 48 and 25 to get 1200.

$$1125a^{2}+1000a+1200=405a^{2}+10125-90\times 45a$$

Multiply -90 and 45 to get -4050.

$$1125a^{2}+1000a+1200=405a^{2}[phone]a$$

Subtract 405a^{2} from both sides.

$$1125a^{2}+1000a+1200-405a^{2}=10125-4050a$$

Combine 1125a^{2} and -405a^{2} to get 720a^{2}.

$$720a^{2}+1000a+1200=10125-4050a$$

Add 4050a to both sides.

$$720a^{2}+1000a+1200+4050a=10125$$

Combine 1000a and 4050a to get 5050a.

$$720a^{2}+5050a+1200=10125$$

Subtract 1200 from both sides.

$$720a^{2}+5050a=10125-1200$$

Subtract 1200 from 10125 to get 8925.

$$720a^{2}+5050a=8925$$

Divide both sides by 720.

$$\frac{720a^{2}+5050a}{720}=\frac{8925}{720}$$

Dividing by 720 undoes the multiplication by 720.

$$a^{2}+\frac{5050}{720}a=\frac{8925}{720}$$

Reduce the fraction \frac{5050}{720} to lowest terms by extracting and canceling out 10.

$$a^{2}+\frac{505}{72}a=\frac{8925}{720}$$

Reduce the fraction \frac{8925}{720} to lowest terms by extracting and canceling out 15.

$$a^{2}+\frac{505}{72}a=\frac{595}{48}$$

Divide \frac{505}{72}, the coefficient of the x term, by 2 to get \frac{505}{144}. Then add the square of \frac{505}{144} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{505}{72}a+\left(\frac{505}{144}\right)^{2}=\frac{595}{48}+\left(\frac{505}{144}\right)^{2}$$

Square \frac{505}{144} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{505}{72}a+\frac{255025}{20736}=\frac{595}{48}+\frac{255025}{20736}$$

Add \frac{595}{48} to \frac{255025}{20736} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{505}{72}a+\frac{255025}{20736}=\frac{512065}{20736}$$

Factor a^{2}+\frac{505}{72}a+\frac{255025}{20736}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{505}{144}\right)^{2}=\frac{512065}{20736}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{505}{144}\right)^{2}}=\sqrt{\frac{512065}{20736}}$$

Simplify.

$$a+\frac{505}{144}=\frac{\sqrt{512065}}{144} a+\frac{505}{144}=-\frac{\sqrt{512065}}{144}$$

Subtract \frac{505}{144} from both sides of the equation.

$$a=\frac{\sqrt{512065}-505}{144} a=\frac{-\sqrt{512065}-505}{144}$$