Solve Using the Quadratic Formula 0.4x-0.7x^2=-2

$0.4 x - 0.7 x^{2} = - 2$

Step 1

Add $2$ to both sides of the equation.

$0.4 x - 0.7 x^{2} + 2 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 0.7$ , $b = 0.4$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 0.4 \pm \sqrt{{0.4}^{2} - 4 \cdot (- 0.7 \cdot 2)}}{2 \cdot - 0.7}$

Step 4

Simplify.

Simplify the numerator.

Raise $0.4$ to the power of $2$ .

$x = \frac{- 0.4 \pm \sqrt{0.16 - 4 \cdot - 0.7 \cdot 2}}{2 \cdot - 0.7}$

Multiply $- 4 \cdot - 0.7 \cdot 2$ .

Multiply $- 4$ by $- 0.7$ .

$x = \frac{- 0.4 \pm \sqrt{0.16 + 2.8 \cdot 2}}{2 \cdot - 0.7}$

Multiply $2.8$ by $2$ .

$x = \frac{- 0.4 \pm \sqrt{0.16 + 5.6}}{2 \cdot - 0.7}$

Add $0.16$ and $5.6$ .

$x = \frac{- 0.4 \pm \sqrt{5.76}}{2 \cdot - 0.7}$

Rewrite $5.76$ as ${2.4}^{2}$ .

$x = \frac{- 0.4 \pm \sqrt{{2.4}^{2}}}{2 \cdot - 0.7}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 0.4 \pm 2.4}{2 \cdot - 0.7}$

Multiply $2$ by $- 0.7$ .

$x = \frac{- 0.4 \pm 2.4}{- 1.4}$

Simplify $\frac{- 0.4 \pm 2.4}{- 1.4}$ .

$x = \frac{2 \pm 12}{7}$

Multiply by $1$ .

$x = \frac{1 (2 \pm 12)}{7}$

Factor $7$ out of $7$ .

$x = \frac{1 (2 \pm 12)}{7 (1)}$

Separate fractions.

$x = \frac{1}{7} \cdot \frac{2 \pm 12}{1}$

Divide $1$ by $7$ .

$x = 0.\overline{142857} (\frac{2 \pm 12}{1})$

Divide $2 \pm 12$ by $1$ .

$x = 0.\overline{142857} (2 \pm 12)$

Step 5

The final answer is the combination of both solutions.

$x = 2, - 1.\overline{428571}$