Find the Roots (Zeros) f(x)=-0.3x^2+x+3

$f (x) = - 0.3 x^{2} + x + 3$

Step 1

Set $- 0.3 x^{2} + x + 3$ equal to $0$ .

$- 0.3 x^{2} + x + 3 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $- 1$ out of $- 0.3 x^{2} + x + 3$ .

Factor $- 1$ out of $- 0.3 x^{2}$ .

$- (0.3 x^{2}) + x + 3 = 0$

Factor $- 1$ out of $x$ .

$- (0.3 x^{2}) - 1 (- x) + 3 = 0$

Rewrite $3$ as $- 1 (- 3)$ .

$- (0.3 x^{2}) - 1 (- x) - 1 \cdot - 3 = 0$

Factor $- 1$ out of $- (0.3 x^{2}) - 1 (- x)$ .

$- (0.3 x^{2} - x) - 1 \cdot - 3 = 0$

Factor $- 1$ out of $- (0.3 x^{2} - x) - 1 (- 3)$ .

$- (0.3 x^{2} - x - 3) = 0$

Factor.

Factor $0.1$ out of $0.3 x^{2} - x - 3$ .

Factor $0.1$ out of $0.3 x^{2}$ .

$- (0.1 (3 x^{2}) - x - 3) = 0$

Factor $0.1$ out of $- x$ .

$- (0.1 (3 x^{2}) + 0.1 (- 10 x) - 3) = 0$

Factor $0.1$ out of $- 3$ .

$- (0.1 (3 x^{2}) + 0.1 (- 10 x) + 0.1 (- 30)) = 0$

Factor $0.1$ out of $0.1 (3 x^{2}) + 0.1 (- 10 x)$ .

$- (0.1 (3 x^{2} - 10 x) + 0.1 (- 30)) = 0$

Factor $0.1$ out of $0.1 (3 x^{2} - 10 x) + 0.1 (- 30)$ .

$- (0.1 (3 x^{2} - 10 x - 30)) = 0$

Remove unnecessary parentheses.

$- 1 \cdot (0.1 (3 x^{2} - 10 x - 30)) = 0$

Multiply $- 1$ by $0.1$ .

$- 0.1 (3 x^{2} - 10 x - 30) = 0$

Divide each term in $- 0.1 (3 x^{2} - 10 x - 30) = 0$ by $- 0.1$ and simplify.

Divide each term in $- 0.1 (3 x^{2} - 10 x - 30) = 0$ by $- 0.1$ .

$\frac{- 0.1 (3 x^{2} - 10 x - 30)}{- 0.1} = \frac{0}{- 0.1}$

Simplify the left side.

Cancel the common factor of $- 0.1$ .

Cancel the common factor.

$\frac{- 0.1 (3 x^{2} - 10 x - 30)}{- 0.1} = \frac{0}{- 0.1}$

Divide $3 x^{2} - 10 x - 30$ by $1$ .

$3 x^{2} - 10 x - 30 = \frac{0}{- 0.1}$

Simplify the right side.

Divide $0$ by $- 0.1$ .

$3 x^{2} - 10 x - 30 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 3$ , $b = - 10$ , and $c = - 30$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (3 \cdot - 30)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot - 30}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 30$ .

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12 \cdot - 30}}{2 \cdot 3}$

Multiply $- 12$ by $- 30$ .

$x = \frac{10 \pm \sqrt{100 + 360}}{2 \cdot 3}$

Add $100$ and $360$ .

$x = \frac{10 \pm \sqrt{460}}{2 \cdot 3}$

Rewrite $460$ as $2^{2} \cdot 115$ .

Factor $4$ out of $460$ .

$x = \frac{10 \pm \sqrt{4 (115)}}{2 \cdot 3}$

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 115}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{115}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{10 \pm 2 \sqrt{115}}{6}$

Simplify $\frac{10 \pm 2 \sqrt{115}}{6}$ .

$x = \frac{5 \pm \sqrt{115}}{3}$

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{115}}{3}, \frac{5 - \sqrt{115}}{3}$

$x = \frac{5 \pm \sqrt{115}}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm \sqrt{115}}{3}$

Decimal Form:

$x = 5.24126843 \dots, - 1.90793509 \dots$

Step 4

Find the Roots (Zeros) f(x)=-0.3x^2+x+3

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