Solve Using the Square Root Property -1/2x^2+1=-39

$- \frac{1}{2} x^{2} + 1 = - 39$

Combine $x^{2}$ and $\frac{1}{2}$ .

$- \frac{x^{2}}{2} + 1 = - 39$

Move all terms not containing $x$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$- \frac{x^{2}}{2} = - 39 - 1$

Subtract $1$ from $- 39$ .

$- \frac{x^{2}}{2} = - 40$

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{x^{2}}{2}) = - 2 \cdot - 40$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{x^{2}}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x^{2}}{2}$ into the numerator.

$- 2 \frac{- x^{2}}{2} = - 2 \cdot - 40$

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- x^{2}}{2} = - 2 \cdot - 40$

Cancel the common factor.

$2 \cdot - 1 \frac{- x^{2}}{2} = - 2 \cdot - 40$

Rewrite the expression.

$- - x^{2} = - 2 \cdot - 40$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - 2 \cdot - 40$

Multiply $x^{2}$ by $1$ .

$x^{2} = - 2 \cdot - 40$

Simplify the right side.

Multiply $- 2$ by $- 40$ .

$x^{2} = 80$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{80}$

Simplify $\pm \sqrt{80}$ .

Rewrite $80$ as $4^{2} \cdot 5$ .

Factor $16$ out of $80$ .

$x = \pm \sqrt{16 (5)}$

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2} \cdot 5}$

Pull terms out from under the radical.

$x = \pm 4 \sqrt{5}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 4 \sqrt{5}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4 \sqrt{5}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4 \sqrt{5}, - 4 \sqrt{5}$

The result can be shown in multiple forms.

Exact Form:

$x = 4 \sqrt{5}, - 4 \sqrt{5}$

Decimal Form:

$x = 8.94427190 \dots, - 8.94427190 \dots$

Solve Using the Square Root Property -1/2x^2+1=-39

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