Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x^2-6=0
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Equation -9x=36
- Express {x} in terms of y where:
- -6*x-11*y=-20
- -15*x+7*y=1
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- -8*x-16*y=-1
- Derivative of:
- x^2-3*x-3
- Factor polynomial:
- x^2-3*x-3
- Factor squared:
- x^2-3*x-3
-
Identical expressions
- x^ two - three *x- three = zero
- x squared minus 3 multiply by x minus 3 equally 0
- x to the power of two minus three multiply by x minus three equally zero
- x2-3*x-3=0
- x²-3*x-3=0
- x to the power of 2-3*x-3=0
- x^2-3x-3=0
- x2-3x-3=0
- x^2-3*x-3=O
-
Similar expressions
- x^2-3*x+3=0
- x^2+3*x-3=0
x^2-3*x-3=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
$$a = 1$$
$$b = -3$$
$$c = -3$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{3}{2} + \frac{\sqrt{21}}{2}$$
$$x_{2} = \frac{3}{2} - \frac{\sqrt{21}}{2}$$
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
$$a = 1$$
$$b = -3$$
$$c = -3$$
, then
D = b^2 - 4 * a * c =
(-3)^2 - 4 * (1) * (-3) = 21
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{3}{2} + \frac{\sqrt{21}}{2}$$
$$x_{2} = \frac{3}{2} - \frac{\sqrt{21}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = -3$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 3$$
$$x_{1} x_{2} = -3$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = -3$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 3$$
$$x_{1} x_{2} = -3$$
Rapid solution
[src]
____
3 \/ 21
x1 = - - ------
2 2
$$x_{1} = \frac{3}{2} - \frac{\sqrt{21}}{2}$$
____
3 \/ 21
x2 = - + ------
2 2
$$x_{2} = \frac{3}{2} + \frac{\sqrt{21}}{2}$$
x2 = 3/2 + sqrt(21)/2
Sum and product of roots
[src]
sum
____ ____ 3 \/ 21 3 \/ 21 - - ------ + - + ------ 2 2 2 2
$$\left(\frac{3}{2} - \frac{\sqrt{21}}{2}\right) + \left(\frac{3}{2} + \frac{\sqrt{21}}{2}\right)$$
=
3
$$3$$
product
/ ____\ / ____\ |3 \/ 21 | |3 \/ 21 | |- - ------|*|- + ------| \2 2 / \2 2 /
$$\left(\frac{3}{2} - \frac{\sqrt{21}}{2}\right) \left(\frac{3}{2} + \frac{\sqrt{21}}{2}\right)$$
=
-3
$$-3$$
-3
The graph