Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 9x^2+6x+1=0
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Equation x^3-7x+6=0
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Equation x^2+5*x+9=0
- Equation x^2+9*x-36=0
- Express {x} in terms of y where:
- 20*x+15*y=1
- 5*x-3*y=8
- 7*x-17*y=14
- -5*x-16*y=9
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Identical expressions
- x^ three -7x+ six = zero
- x cubed minus 7x plus 6 equally 0
- x to the power of three minus 7x plus six equally zero
- x3-7x+6=0
- x³-7x+6=0
- x to the power of 3-7x+6=0
- x^3-7x+6=O
-
Similar expressions
- x^3+7x+6=0
- x^3-7x-6=0
x^3-7x+6=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(x^{3} - 7 x\right) + 6 = 0$$
transform
$$\left(- 7 x + \left(x^{3} - 1\right)\right) + 7 = 0$$
or
$$\left(- 7 x + \left(x^{3} - 1^{3}\right)\right) + 7 = 0$$
$$- 7 \left(x - 1\right) + \left(x^{3} - 1^{3}\right) = 0$$
$$\left(x - 1\right) \left(\left(x^{2} + x\right) + 1^{2}\right) - 7 \left(x - 1\right) = 0$$
Take common factor -1 + x from the equation
we get:
$$\left(x - 1\right) \left(\left(\left(x^{2} + x\right) + 1^{2}\right) - 7\right) = 0$$
or
$$\left(x - 1\right) \left(x^{2} + x - 6\right) = 0$$
then:
$$x_{1} = 1$$
and also
we get the equation
$$x^{2} + x - 6 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 1$$
$$c = -6$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{2} = 2$$
$$x_{3} = -3$$
The final answer for x^3 - 7*x + 6 = 0:
$$x_{1} = 1$$
$$x_{2} = 2$$
$$x_{3} = -3$$
$$\left(x^{3} - 7 x\right) + 6 = 0$$
transform
$$\left(- 7 x + \left(x^{3} - 1\right)\right) + 7 = 0$$
or
$$\left(- 7 x + \left(x^{3} - 1^{3}\right)\right) + 7 = 0$$
$$- 7 \left(x - 1\right) + \left(x^{3} - 1^{3}\right) = 0$$
$$\left(x - 1\right) \left(\left(x^{2} + x\right) + 1^{2}\right) - 7 \left(x - 1\right) = 0$$
Take common factor -1 + x from the equation
we get:
$$\left(x - 1\right) \left(\left(\left(x^{2} + x\right) + 1^{2}\right) - 7\right) = 0$$
or
$$\left(x - 1\right) \left(x^{2} + x - 6\right) = 0$$
then:
$$x_{1} = 1$$
and also
we get the equation
$$x^{2} + x - 6 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 1$$
$$c = -6$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (1) * (-6) = 25
Because D > 0, then the equation has two roots.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = 2$$
$$x_{3} = -3$$
The final answer for x^3 - 7*x + 6 = 0:
$$x_{1} = 1$$
$$x_{2} = 2$$
$$x_{3} = -3$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -7$$
$$v = \frac{d}{a}$$
$$v = 6$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -7$$
$$x_{1} x_{2} x_{3} = 6$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -7$$
$$v = \frac{d}{a}$$
$$v = 6$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -7$$
$$x_{1} x_{2} x_{3} = 6$$
Sum and product of roots
[src]
sum
-3 + 1 + 2
$$\left(-3 + 1\right) + 2$$
=
0
$$0$$
product
-3*2
$$- 6$$
=
-6
$$-6$$
-6
The graph