Find the Roots (Zeros) f(x)=1/2x^2+5/2x-3/2

$f (x) = \frac{1}{2} x^{2} + \frac{5}{2} x - \frac{3}{2}$

Set $\frac{1}{2} x^{2} + \frac{5}{2} x - \frac{3}{2}$ equal to $0$ .

$\frac{1}{2} x^{2} + \frac{5}{2} x - \frac{3}{2} = 0$

Solve for $x$ .

Simplify each term.

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

$\frac{x^{2}}{2} + \frac{5}{2} x - \frac{3}{2} = 0$

Combine $\frac{5}{2}$ and $x$ .

$\frac{x^{2}}{2} + \frac{5 x}{2} - \frac{3}{2} = 0$

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

Apply the distributive property.

$2 (\frac{x^{2}}{2}) + 2 (\frac{5 x}{2}) + 2 (- \frac{3}{2}) = 0$

Simplify.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 (\frac{x^{2}}{2}) + 2 (\frac{5 x}{2}) + 2 (- \frac{3}{2}) = 0$

Rewrite the expression.

$x^{2} + 2 (\frac{5 x}{2}) + 2 (- \frac{3}{2}) = 0$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2} + 2 (\frac{5 x}{2}) + 2 (- \frac{3}{2}) = 0$

Rewrite the expression.

$x^{2} + 5 x + 2 (- \frac{3}{2}) = 0$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$x^{2} + 5 x + 2 (\frac{- 3}{2}) = 0$

Cancel the common factor.

$x^{2} + 5 x + 2 (\frac{- 3}{2}) = 0$

Rewrite the expression.

$x^{2} + 5 x - 3 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 5$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 3)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 3}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 3}}{2 \cdot 1}$

Multiply $- 4$ by $- 3$ .

$x = \frac{- 5 \pm \sqrt{25 + 12}}{2 \cdot 1}$

Add $25$ and $12$ .

$x = \frac{- 5 \pm \sqrt{37}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{37}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{5 - \sqrt{37}}{2}, - \frac{5 + \sqrt{37}}{2}$

$x = \frac{- 5 \pm \sqrt{37}}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 5 \pm \sqrt{37}}{2}$

Decimal Form:

$x = 0.54138126 \dots, - 5.54138126 \dots$

Find the Roots (Zeros) f(x)=1/2x^2+5/2x-3/2

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