Solve 1/2x^2-16=34

$$\frac { 1 } { 2 } x ^ { 2 } - 16 = 34$$

Solve for x

$$x=10$$

$$x=-10$$

Steps Using Difference of Squares

Subtract 34 from both sides.

$$\frac{1}{2}x^{2}-16-34=0$$

Subtract 34 from -16 to get -50.

$$\frac{1}{2}x^{2}-50=0$$

Multiply both sides by 2.

$$x^{2}-100=0$$

Consider x^{2}-100. Rewrite x^{2}-100 as x^{2}-10^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-10\right)\left(x+10\right)=0$$

To find equation solutions, solve x-10=0 and x+10=0.

$$x=10 x=-10$$

Steps by Finding Square Root

Add 16 to both sides.

$$\frac{1}{2}x^{2}=34+16$$

Add 34 and 16 to get 50.

$$\frac{1}{2}x^{2}=50$$

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$x^{2}=50\times 2$$

Multiply 50 and 2 to get 100.

$$x^{2}=100$$

Take the square root of both sides of the equation.

$$x=10 x=-10$$

Steps Using the Quadratic Formula

Subtract 34 from both sides.

$$\frac{1}{2}x^{2}-16-34=0$$

Subtract 34 from -16 to get -50.

$$\frac{1}{2}x^{2}-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 0 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{1}{2}\left(-50\right)}}{2\times \frac{1}{2}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{1}{2}\left(-50\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x=\frac{0±\sqrt{-2\left(-50\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -50.

$$x=\frac{0±\sqrt{100}}{2\times \frac{1}{2}}$$

Take the square root of 100.

$$x=\frac{0±10}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{0±10}{1}$$

Now solve the equation x=\frac{0±10}{1} when ± is plus.

$$x=10$$

Now solve the equation x=\frac{0±10}{1} when ± is minus.

$$x=-10$$

The equation is now solved.

$$x=10 x=-10$$