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23x-10+5x^2=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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