Solve 2.25a^2+9a^2=52

$$2.25 a ^ { 2 } + 9 a ^ { 2 } = 52$$

Solve for a

$$a = \frac{4 \sqrt{65}}{15} \approx 2.1499354$$

$$a = -\frac{4 \sqrt{65}}{15} \approx -2.1499354$$

Steps by Finding Square Root

Combine 2.25a^{2} and 9a^{2} to get 11.25a^{2}.

$$11.25a^{2}=52$$

Divide both sides by 11.25.

$$a^{2}=\frac{52}{11.25}$$

Expand \frac{52}{11.25} by multiplying both numerator and the denominator by 100.

$$a^{2}=\frac{5200}{1125}$$

Reduce the fraction \frac{5200}{1125} to lowest terms by extracting and canceling out 25.

$$a^{2}=\frac{208}{45}$$

Take the square root of both sides of the equation.

$$a=\frac{4\sqrt{65}}{15} a=-\frac{4\sqrt{65}}{15}$$

Steps Using the Quadratic Formula

Combine 2.25a^{2} and 9a^{2} to get 11.25a^{2}.

$$11.25a^{2}=52$$

Subtract 52 from both sides.

$$11.25a^{2}-52=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 11.25 for a, 0 for b, and -52 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{0±\sqrt{0^{2}-4\times 11.25\left(-52\right)}}{2\times 11.25}$$

Square 0.

$$a=\frac{0±\sqrt{-4\times 11.25\left(-52\right)}}{2\times 11.25}$$

Multiply -4 times 11.25.

$$a=\frac{0±\sqrt{-45\left(-52\right)}}{2\times 11.25}$$

Multiply -45 times -52.

$$a=\frac{0±\sqrt{2340}}{2\times 11.25}$$

Take the square root of 2340.

$$a=\frac{0±6\sqrt{65}}{2\times 11.25}$$

Multiply 2 times 11.25.

$$a=\frac{0±6\sqrt{65}}{22.5}$$

Now solve the equation a=\frac{0±6\sqrt{65}}{22.5} when ± is plus.

$$a=\frac{4\sqrt{65}}{15}$$

Now solve the equation a=\frac{0±6\sqrt{65}}{22.5} when ± is minus.

$$a=-\frac{4\sqrt{65}}{15}$$

The equation is now solved.

$$a=\frac{4\sqrt{65}}{15} a=-\frac{4\sqrt{65}}{15}$$