Mister Exam
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How to use it?
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Similar expressions
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2x^2+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
$$a = 2$$
$$b = 5$$
$$c = -2$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{5}{4} + \frac{\sqrt{41}}{4}$$
$$x_{2} = - \frac{\sqrt{41}}{4} - \frac{5}{4}$$
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
$$a = 2$$
$$b = 5$$
$$c = -2$$
, then
D = b^2 - 4 * a * c =
(5)^2 - 4 * (2) * (-2) = 41
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{5}{4} + \frac{\sqrt{41}}{4}$$
$$x_{2} = - \frac{\sqrt{41}}{4} - \frac{5}{4}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x^{2} + 5 x\right) - 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{5 x}{2} - 1 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{5}{2}$$
$$q = \frac{c}{a}$$
$$q = -1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{5}{2}$$
$$x_{1} x_{2} = -1$$
$$\left(2 x^{2} + 5 x\right) - 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{5 x}{2} - 1 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{5}{2}$$
$$q = \frac{c}{a}$$
$$q = -1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{5}{2}$$
$$x_{1} x_{2} = -1$$
Rapid solution
[src]
____
5 \/ 41
x1 = - - + ------
4 4
$$x_{1} = - \frac{5}{4} + \frac{\sqrt{41}}{4}$$
____
5 \/ 41
x2 = - - - ------
4 4
$$x_{2} = - \frac{\sqrt{41}}{4} - \frac{5}{4}$$
x2 = -sqrt(41)/4 - 5/4
Sum and product of roots
[src]
sum
____ ____ 5 \/ 41 5 \/ 41 - - + ------ + - - - ------ 4 4 4 4
$$\left(- \frac{\sqrt{41}}{4} - \frac{5}{4}\right) + \left(- \frac{5}{4} + \frac{\sqrt{41}}{4}\right)$$
=
-5/2
$$- \frac{5}{2}$$
product
/ ____\ / ____\ | 5 \/ 41 | | 5 \/ 41 | |- - + ------|*|- - - ------| \ 4 4 / \ 4 4 /
$$\left(- \frac{5}{4} + \frac{\sqrt{41}}{4}\right) \left(- \frac{\sqrt{41}}{4} - \frac{5}{4}\right)$$
=
-1
$$-1$$
-1
The graph