Solve 50-1/2x^2=23

$$50 - \frac { 1 } { 2 } x ^ { 2 } = 23$$

Solve for x

$$x=3\sqrt{6}\approx 7.348469228$$

$$x=-3\sqrt{6}\approx -7.348469228$$

Steps by Finding Square Root

Subtract 50 from both sides.

$$-\frac{1}{2}x^{2}=23-50$$

Subtract 50 from 23 to get -27.

$$-\frac{1}{2}x^{2}=-27$$

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

$$x^{2}=-27\left(-2\right)$$

Multiply -27 and -2 to get 54.

$$x^{2}=54$$

Take the square root of both sides of the equation.

$$x=3\sqrt{6} x=-3\sqrt{6}$$

Steps Using the Quadratic Formula

Subtract 23 from both sides.

$$50-\frac{1}{2}x^{2}-23=0$$

Subtract 23 from 50 to get 27.

$$27-\frac{1}{2}x^{2}=0$$

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.

$$-\frac{1}{2}x^{2}+27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{2} for a, 0 for b, and 27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{1}{2}\right)\times 27}}{2\left(-\frac{1}{2}\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{1}{2}\right)\times 27}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{0±\sqrt{2\times 27}}{2\left(-\frac{1}{2}\right)}$$

Multiply 2 times 27.

$$x=\frac{0±\sqrt{54}}{2\left(-\frac{1}{2}\right)}$$

Take the square root of 54.

$$x=\frac{0±3\sqrt{6}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{0±3\sqrt{6}}{-1}$$

Now solve the equation x=\frac{0±3\sqrt{6}}{-1} when ± is plus.

$$x=-3\sqrt{6}$$

Now solve the equation x=\frac{0±3\sqrt{6}}{-1} when ± is minus.

$$x=3\sqrt{6}$$

The equation is now solved.

$$x=-3\sqrt{6} x=3\sqrt{6}$$