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Detail solution

This equation is of the form

A quadratic equation can be solved

using the discriminant.

The roots of the quadratic equation:

$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$

$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$

where D = b^2 - 4*a*c - it is the discriminant.

Because

$$a = 5$$

$$b = 23$$

$$c = -10$$

, then

Because D > 0, then the equation has two roots.

or

$$x_{1} = \frac{2}{5}$$

$$x_{2} = -5$$

a*x^2 + b*x + c = 0

A quadratic equation can be solved

using the discriminant.

The roots of the quadratic equation:

$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$

$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$

where D = b^2 - 4*a*c - it is the discriminant.

Because

$$a = 5$$

$$b = 23$$

$$c = -10$$

, then

D = b^2 - 4 * a * c =

(23)^2 - 4 * (5) * (-10) = 729

Because D > 0, then the equation has two roots.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or

$$x_{1} = \frac{2}{5}$$

$$x_{2} = -5$$

Vieta's Theorem

rewrite the equation

$$5 x^{2} + \left(23 x - 10\right) = 0$$

of

$$a x^{2} + b x + c = 0$$

as reduced quadratic equation

$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$

$$x^{2} + \frac{23 x}{5} - 2 = 0$$

$$p x + q + x^{2} = 0$$

where

$$p = \frac{b}{a}$$

$$p = \frac{23}{5}$$

$$q = \frac{c}{a}$$

$$q = -2$$

Vieta Formulas

$$x_{1} + x_{2} = - p$$

$$x_{1} x_{2} = q$$

$$x_{1} + x_{2} = - \frac{23}{5}$$

$$x_{1} x_{2} = -2$$

$$5 x^{2} + \left(23 x - 10\right) = 0$$

of

$$a x^{2} + b x + c = 0$$

as reduced quadratic equation

$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$

$$x^{2} + \frac{23 x}{5} - 2 = 0$$

$$p x + q + x^{2} = 0$$

where

$$p = \frac{b}{a}$$

$$p = \frac{23}{5}$$

$$q = \frac{c}{a}$$

$$q = -2$$

Vieta Formulas

$$x_{1} + x_{2} = - p$$

$$x_{1} x_{2} = q$$

$$x_{1} + x_{2} = - \frac{23}{5}$$

$$x_{1} x_{2} = -2$$

Sum and product of roots
[src]

sum

-5 + 2/5

$$-5 + \frac{2}{5}$$

=

-23/5

$$- \frac{23}{5}$$

product

-5*2 ---- 5

$$- 2$$

=

-2

$$-2$$

-2

The graph