Mister Exam
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Similar expressions
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3x^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
$$a = 3$$
$$b = -5$$
$$c = -2$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 2$$
Simplify
$$x_{2} = - \frac{1}{3}$$
Simplify
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
$$a = 3$$
$$b = -5$$
$$c = -2$$
, then
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (3) * (-2) = 49
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 2$$
Simplify
$$x_{2} = - \frac{1}{3}$$
Simplify
Vieta's Theorem
rewrite the equation
$$3 x^{2} - 5 x - 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{5 x}{3} - \frac{2}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{5}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{5}{3}$$
$$x_{1} x_{2} = - \frac{2}{3}$$
$$3 x^{2} - 5 x - 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{5 x}{3} - \frac{2}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{5}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{5}{3}$$
$$x_{1} x_{2} = - \frac{2}{3}$$
Sum and product of roots
[src]
sum
0 - 1/3 + 2
$$\left(- \frac{1}{3} + 0\right) + 2$$
=
5/3
$$\frac{5}{3}$$
product
1*-1/3*2
$$1 \left(- \frac{1}{3}\right) 2$$
=
-2/3
$$- \frac{2}{3}$$
-2/3
The graph