Solve 14-9a^2=-16-4a^2

$$14 - 9 a ^ { 2 } = - 16 - 4 a ^ { 2 }$$

Solve for a

$$a=\sqrt{6}\approx 2.449489743$$

$$a=-\sqrt{6}\approx -2.449489743$$

Steps by Finding Square Root

Add 4a^{2} to both sides.

$$14-9a^{2}+4a^{2}=-16$$

Combine -9a^{2} and 4a^{2} to get -5a^{2}.

$$14-5a^{2}=-16$$

Subtract 14 from both sides.

$$-5a^{2}=-16-14$$

Subtract 14 from -16 to get -30.

$$-5a^{2}=-30$$

Divide both sides by -5.

$$a^{2}=\frac{-30}{-5}$$

Divide -30 by -5 to get 6.

$$a^{2}=6$$

Take the square root of both sides of the equation.

$$a=\sqrt{6} a=-\sqrt{6}$$

Steps Using the Quadratic Formula

Subtract -16 from both sides.

$$14-9a^{2}-\left(-16\right)=-4a^{2}$$

The opposite of -16 is 16.

$$14-9a^{2}+16=-4a^{2}$$

Add 4a^{2} to both sides.

$$14-9a^{2}+16+4a^{2}=0$$

Add 14 and 16 to get 30.

$$30-9a^{2}+4a^{2}=0$$

Combine -9a^{2} and 4a^{2} to get -5a^{2}.

$$30-5a^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-5a^{2}+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 0 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{0±\sqrt{0^{2}-4\left(-5\right)\times 30}}{2\left(-5\right)}$$

Square 0.

$$a=\frac{0±\sqrt{-4\left(-5\right)\times 30}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$a=\frac{0±\sqrt{20\times 30}}{2\left(-5\right)}$$

Multiply 20 times 30.

$$a=\frac{0±\sqrt{600}}{2\left(-5\right)}$$

Take the square root of 600.

$$a=\frac{0±10\sqrt{6}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$a=\frac{0±10\sqrt{6}}{-10}$$

Now solve the equation a=\frac{0±10\sqrt{6}}{-10} when ± is plus.

$$a=-\sqrt{6}$$

Now solve the equation a=\frac{0±10\sqrt{6}}{-10} when ± is minus.

$$a=\sqrt{6}$$

The equation is now solved.

$$a=-\sqrt{6} a=\sqrt{6}$$