Solve 15x^2-24=1-x^2

$$15 x ^ { 2 } - 24 = 1 - x ^ { 2 }$$

Solve for x

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

$$x = -\frac{5}{4} = -1\frac{1}{4} = -1.25$$

Steps Using Difference of Squares

Subtract 1 from both sides.

$$15x^{2}-24-1=-x^{2}$$

Subtract 1 from -24 to get -25.

$$15x^{2}-25=-x^{2}$$

Add x^{2} to both sides.

$$15x^{2}-25+x^{2}=0$$

Combine 15x^{2} and x^{2} to get 16x^{2}.

$$16x^{2}-25=0$$

Consider 16x^{2}-25. Rewrite 16x^{2}-25 as \left(4x\right)^{2}-5^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(4x-5\right)\left(4x+5\right)=0$$

To find equation solutions, solve 4x-5=0 and 4x+5=0.

$$x=\frac{5}{4} x=-\frac{5}{4}$$

Steps by Finding Square Root

Add x^{2} to both sides.

$$15x^{2}-24+x^{2}=1$$

Combine 15x^{2} and x^{2} to get 16x^{2}.

$$16x^{2}-24=1$$

Add 24 to both sides.

$$16x^{2}=1+24$$

Add 1 and 24 to get 25.

$$16x^{2}=25$$

Divide both sides by 16.

$$x^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$x=\frac{5}{4} x=-\frac{5}{4}$$

Steps Using the Quadratic Formula

Subtract 1 from both sides.

$$15x^{2}-24-1=-x^{2}$$

Subtract 1 from -24 to get -25.

$$15x^{2}-25=-x^{2}$$

Add x^{2} to both sides.

$$15x^{2}-25+x^{2}=0$$

Combine 15x^{2} and x^{2} to get 16x^{2}.

$$16x^{2}-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 0 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 16\left(-25\right)}}{2\times 16}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 16\left(-25\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{0±\sqrt{-64\left(-25\right)}}{2\times 16}$$

Multiply -64 times -25.

$$x=\frac{0±\sqrt{1600}}{2\times 16}$$

Take the square root of 1600.

$$x=\frac{0±40}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{0±40}{32}$$

Now solve the equation x=\frac{0±40}{32} when ± is plus. Reduce the fraction \frac{40}{32} to lowest terms by extracting and canceling out 8.

$$x=\frac{5}{4}$$

Now solve the equation x=\frac{0±40}{32} when ± is minus. Reduce the fraction \frac{-40}{32} to lowest terms by extracting and canceling out 8.

$$x=-\frac{5}{4}$$

The equation is now solved.

$$x=\frac{5}{4} x=-\frac{5}{4}$$