Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x^2+7x=18
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Equation x^2-4x=0
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Equation (y^4+4*y^2-12)/4=0
-
Equation x^2+7x-18=0
- Express {x} in terms of y where:
- -2*x-19*y=3
- -2*x+18*y=13
- -2*x-15*y=-1
- 10*x-2*y=11
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Identical expressions
- x^ two +x+ thirty = zero
- x squared plus x plus 30 equally 0
- x to the power of two plus x plus thirty equally zero
- x2+x+30=0
- x²+x+30=0
- x to the power of 2+x+30=0
- x^2+x+30=O
-
Similar expressions
- x^2+x-30=0
- x^2-x+30=0
x^2+x+30=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
$$a = 1$$
$$b = 1$$
$$c = 30$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{119} i}{2}$$
$$x_{2} = - \frac{1}{2} - \frac{\sqrt{119} i}{2}$$
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
$$a = 1$$
$$b = 1$$
$$c = 30$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (1) * (30) = -119
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{119} i}{2}$$
$$x_{2} = - \frac{1}{2} - \frac{\sqrt{119} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = 30$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = 30$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = 30$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = 30$$
Sum and product of roots
[src]
sum
_____ _____ 1 I*\/ 119 1 I*\/ 119 - - - --------- + - - + --------- 2 2 2 2
$$\left(- \frac{1}{2} - \frac{\sqrt{119} i}{2}\right) + \left(- \frac{1}{2} + \frac{\sqrt{119} i}{2}\right)$$
=
-1
$$-1$$
product
/ _____\ / _____\ | 1 I*\/ 119 | | 1 I*\/ 119 | |- - - ---------|*|- - + ---------| \ 2 2 / \ 2 2 /
$$\left(- \frac{1}{2} - \frac{\sqrt{119} i}{2}\right) \left(- \frac{1}{2} + \frac{\sqrt{119} i}{2}\right)$$
=
30
$$30$$
30
Rapid solution
[src]
_____
1 I*\/ 119
x1 = - - - ---------
2 2
$$x_{1} = - \frac{1}{2} - \frac{\sqrt{119} i}{2}$$
_____
1 I*\/ 119
x2 = - - + ---------
2 2
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{119} i}{2}$$
x2 = -1/2 + sqrt(119)*i/2
Numerical answer
[src]
x1 = -0.5 - 5.45435605731786*i
x2 = -0.5 + 5.45435605731786*i
x2 = -0.5 + 5.45435605731786*i