Solve Using the Quadratic Formula (x^2)/2+7/2x=2

$\frac{x^{2}}{2} + \frac{7}{2} x = 2$

Step 1

Move all terms to the left side of the equation and simplify.

Simplify the left side.

Combine $\frac{7}{2}$ and $x$ .

$\frac{x^{2}}{2} + \frac{7 x}{2} = 2$

Subtract $2$ from both sides of the equation.

$\frac{x^{2}}{2} + \frac{7 x}{2} - 2 = 0$

Step 2

Multiply through by the least common denominator $2$ , then simplify.

Apply the distributive property.

$2 (\frac{x^{2}}{2}) + 2 (\frac{7 x}{2}) + 2 \cdot - 2 = 0$

Simplify.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 (\frac{x^{2}}{2}) + 2 (\frac{7 x}{2}) + 2 \cdot - 2 = 0$

Rewrite the expression.

$x^{2} + 2 (\frac{7 x}{2}) + 2 \cdot - 2 = 0$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2} + 2 (\frac{7 x}{2}) + 2 \cdot - 2 = 0$

Rewrite the expression.

$x^{2} + 7 x + 2 \cdot - 2 = 0$

Multiply $2$ by $- 2$ .

$x^{2} + 7 x - 4 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = 7$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (1 \cdot - 4)}}{2 \cdot 1}$

Step 5

Simplify.

Simplify the numerator.

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 4$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 4}}{2 \cdot 1}$

Multiply $- 4$ by $- 4$ .

$x = \frac{- 7 \pm \sqrt{49 + 16}}{2 \cdot 1}$

Add $49$ and $16$ .

$x = \frac{- 7 \pm \sqrt{65}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{65}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 7 \pm \sqrt{65}}{2}$

Decimal Form:

$x = 0.53112887 \dots, - 7.53112887 \dots$

Solve Using the Quadratic Formula (x^2)/2+7/2x=2

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