Mister Exam
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How to use it?
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Identical expressions
- 3x²+67x- one = zero
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Similar expressions
- 3x²-67x-1=0
- 3x²+67x+1=0
3x²+67x-1=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
$$a = 3$$
$$b = 67$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{67}{6} + \frac{\sqrt{4501}}{6}$$
$$x_{2} = - \frac{\sqrt{4501}}{6} - \frac{67}{6}$$
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
$$a = 3$$
$$b = 67$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(67)^2 - 4 * (3) * (-1) = 4501
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{67}{6} + \frac{\sqrt{4501}}{6}$$
$$x_{2} = - \frac{\sqrt{4501}}{6} - \frac{67}{6}$$
Vieta's Theorem
rewrite the equation
$$\left(3 x^{2} + 67 x\right) - 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{67 x}{3} - \frac{1}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{67}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{67}{3}$$
$$x_{1} x_{2} = - \frac{1}{3}$$
$$\left(3 x^{2} + 67 x\right) - 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{67 x}{3} - \frac{1}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{67}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{67}{3}$$
$$x_{1} x_{2} = - \frac{1}{3}$$
Sum and product of roots
[src]
sum
______ ______ 67 \/ 4501 67 \/ 4501 - -- + -------- + - -- - -------- 6 6 6 6
$$\left(- \frac{\sqrt{4501}}{6} - \frac{67}{6}\right) + \left(- \frac{67}{6} + \frac{\sqrt{4501}}{6}\right)$$
=
-67/3
$$- \frac{67}{3}$$
product
/ ______\ / ______\ | 67 \/ 4501 | | 67 \/ 4501 | |- -- + --------|*|- -- - --------| \ 6 6 / \ 6 6 /
$$\left(- \frac{67}{6} + \frac{\sqrt{4501}}{6}\right) \left(- \frac{\sqrt{4501}}{6} - \frac{67}{6}\right)$$
=
-1/3
$$- \frac{1}{3}$$
-1/3
Rapid solution
[src]
______
67 \/ 4501
x1 = - -- + --------
6 6
$$x_{1} = - \frac{67}{6} + \frac{\sqrt{4501}}{6}$$
______
67 \/ 4501
x2 = - -- - --------
6 6
$$x_{2} = - \frac{\sqrt{4501}}{6} - \frac{67}{6}$$
x2 = -sqrt(4501)/6 - 67/6