Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 16x-3=8x-43
-
Equation 2*sin(x)=1
-
Equation 3^x=-1
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Equation x/2+x=12
- Express {x} in terms of y where:
- 10*x-6*y=9
- -11*x-11*y=4
- 16*x+14*y=17
- 9*x+8*y=-4
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Identical expressions
- x^ two -5x+ twenty = zero
- x squared minus 5x plus 20 equally 0
- x to the power of two minus 5x plus twenty equally zero
- x2-5x+20=0
- x²-5x+20=0
- x to the power of 2-5x+20=0
- x^2-5x+20=O
-
Similar expressions
- x^2+5x+20=0
- x^2-5x-20=0
x^2-5x+20=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
$$a = 1$$
$$b = -5$$
$$c = 20$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{55} i}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{55} i}{2}$$
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
$$a = 1$$
$$b = -5$$
$$c = 20$$
, then
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (1) * (20) = -55
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
$$x_{1} = \frac{5}{2} + \frac{\sqrt{55} i}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{55} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 20$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = 20$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 20$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = 20$$
Rapid solution
[src]
____
5 I*\/ 55
x1 = - - --------
2 2
$$x_{1} = \frac{5}{2} - \frac{\sqrt{55} i}{2}$$
____
5 I*\/ 55
x2 = - + --------
2 2
$$x_{2} = \frac{5}{2} + \frac{\sqrt{55} i}{2}$$
x2 = 5/2 + sqrt(55)*i/2
Sum and product of roots
[src]
sum
____ ____ 5 I*\/ 55 5 I*\/ 55 - - -------- + - + -------- 2 2 2 2
$$\left(\frac{5}{2} - \frac{\sqrt{55} i}{2}\right) + \left(\frac{5}{2} + \frac{\sqrt{55} i}{2}\right)$$
=
5
$$5$$
product
/ ____\ / ____\ |5 I*\/ 55 | |5 I*\/ 55 | |- - --------|*|- + --------| \2 2 / \2 2 /
$$\left(\frac{5}{2} - \frac{\sqrt{55} i}{2}\right) \left(\frac{5}{2} + \frac{\sqrt{55} i}{2}\right)$$
=
20
$$20$$
20
Numerical answer
[src]
x1 = 2.5 - 3.70809924354783*i
x2 = 2.5 + 3.70809924354783*i
x2 = 2.5 + 3.70809924354783*i