Solve 9+x^2=14-x^2

$$9 + x ^ { 2 } = 14 - x ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{10}}{2} \approx 1.58113883$$

$$x = -\frac{\sqrt{10}}{2} \approx -1.58113883$$

Steps by Finding Square Root

Add x^{2} to both sides.

$$9+x^{2}+x^{2}=14$$

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

$$9+2x^{2}=14$$

Subtract 9 from both sides.

$$2x^{2}=14-9$$

Subtract 9 from 14 to get 5.

$$2x^{2}=5$$

Divide both sides by 2.

$$x^{2}=\frac{5}{2}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{10}}{2} x=-\frac{\sqrt{10}}{2}$$

Steps Using the Quadratic Formula

Subtract 14 from both sides.

$$9+x^{2}-14=-x^{2}$$

Subtract 14 from 9 to get -5.

$$-5+x^{2}=-x^{2}$$

Add x^{2} to both sides.

$$-5+x^{2}+x^{2}=0$$

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

$$-5+2x^{2}=0$$

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.

$$2x^{2}-5=0$$

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

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

Square 0.

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

Multiply -4 times 2.

$$x=\frac{0±\sqrt{-8\left(-5\right)}}{2\times 2}$$

Multiply -8 times -5.

$$x=\frac{0±\sqrt{40}}{2\times 2}$$

Take the square root of 40.

$$x=\frac{0±2\sqrt{10}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{0±2\sqrt{10}}{4}$$

Now solve the equation x=\frac{0±2\sqrt{10}}{4} when ± is plus.

$$x=\frac{\sqrt{10}}{2}$$

Now solve the equation x=\frac{0±2\sqrt{10}}{4} when ± is minus.

$$x=-\frac{\sqrt{10}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{10}}{2} x=-\frac{\sqrt{10}}{2}$$