Solve 36n^2-24=-15

$$36 n ^ { 2 } - 24 = - 15$$

Solve for n

$$n=\frac{1}{2}=0.5$$

$$n=-\frac{1}{2}=-0.5$$

Steps Using Difference of Squares

Add 15 to both sides.

$$36n^{2}-24+15=0$$

Add -24 and 15 to get -9.

$$36n^{2}-9=0$$

Divide both sides by 9.

$$4n^{2}-1=0$$

Consider 4n^{2}-1. Rewrite 4n^{2}-1 as \left(2n\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2n-1\right)\left(2n+1\right)=0$$

To find equation solutions, solve 2n-1=0 and 2n+1=0.

$$n=\frac{1}{2} n=-\frac{1}{2}$$

Steps by Finding Square Root

Add 24 to both sides.

$$36n^{2}=-15+24$$

Add -15 and 24 to get 9.

$$36n^{2}=9$$

Divide both sides by 36.

$$n^{2}=\frac{9}{36}$$

Reduce the fraction \frac{9}{36} to lowest terms by extracting and canceling out 9.

$$n^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$n=\frac{1}{2} n=-\frac{1}{2}$$

Steps Using the Quadratic Formula

Add 15 to both sides.

$$36n^{2}-24+15=0$$

Add -24 and 15 to get -9.

$$36n^{2}-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, 0 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\times 36\left(-9\right)}}{2\times 36}$$

Square 0.

$$n=\frac{0±\sqrt{-4\times 36\left(-9\right)}}{2\times 36}$$

Multiply -4 times 36.

$$n=\frac{0±\sqrt{-144\left(-9\right)}}{2\times 36}$$

Multiply -144 times -9.

$$n=\frac{0±\sqrt{1296}}{2\times 36}$$

Take the square root of 1296.

$$n=\frac{0±36}{2\times 36}$$

Multiply 2 times 36.

$$n=\frac{0±36}{72}$$

Now solve the equation n=\frac{0±36}{72} when ± is plus. Reduce the fraction \frac{36}{72} to lowest terms by extracting and canceling out 36.

$$n=\frac{1}{2}$$

Now solve the equation n=\frac{0±36}{72} when ± is minus. Reduce the fraction \frac{-36}{72} to lowest terms by extracting and canceling out 36.

$$n=-\frac{1}{2}$$

The equation is now solved.

$$n=\frac{1}{2} n=-\frac{1}{2}$$