Mister Exam
Other calculators
-
How to use it?
- Equation:
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Equation x^2=49
-
Equation cos(pi*(x-1)/3)=1/2
-
Equation x^2-2x-1=0
-
Equation 3*x^2-3=0
- Express {x} in terms of y where:
- 18*x-14*y=-12
- 4*x-18*y=-14
- -14*x-1*y=3
- 10*x-18*y=6
-
Identical expressions
- x^ two -5x+ fourteen = zero
- x squared minus 5x plus 14 equally 0
- x to the power of two minus 5x plus fourteen equally zero
- x2-5x+14=0
- x²-5x+14=0
- x to the power of 2-5x+14=0
- x^2-5x+14=O
-
Similar expressions
- x^2+5x+14=0
- x^2-5x-14=0
x^2-5x+14=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
$$a = 1$$
$$b = -5$$
$$c = 14$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{31} i}{2}$$
Simplify
$$x_{2} = \frac{5}{2} - \frac{\sqrt{31} i}{2}$$
Simplify
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
$$a = 1$$
$$b = -5$$
$$c = 14$$
, then
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (1) * (14) = -31
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
$$x_{1} = \frac{5}{2} + \frac{\sqrt{31} i}{2}$$
Simplify
$$x_{2} = \frac{5}{2} - \frac{\sqrt{31} i}{2}$$
Simplify
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 14$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = 14$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 14$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = 14$$
Rapid solution
[src]
____
5 I*\/ 31
x1 = - - --------
2 2
$$x_{1} = \frac{5}{2} - \frac{\sqrt{31} i}{2}$$
____
5 I*\/ 31
x2 = - + --------
2 2
$$x_{2} = \frac{5}{2} + \frac{\sqrt{31} i}{2}$$
Sum and product of roots
[src]
sum
____ ____
5 I*\/ 31 5 I*\/ 31
0 + - - -------- + - + --------
2 2 2 2
$$\left(0 + \left(\frac{5}{2} - \frac{\sqrt{31} i}{2}\right)\right) + \left(\frac{5}{2} + \frac{\sqrt{31} i}{2}\right)$$
=
5
$$5$$
product
/ ____\ / ____\ |5 I*\/ 31 | |5 I*\/ 31 | 1*|- - --------|*|- + --------| \2 2 / \2 2 /
$$1 \cdot \left(\frac{5}{2} - \frac{\sqrt{31} i}{2}\right) \left(\frac{5}{2} + \frac{\sqrt{31} i}{2}\right)$$
=
14
$$14$$
14
Numerical answer
[src]
x1 = 2.5 + 2.78388218141501*i
x2 = 2.5 - 2.78388218141501*i
x2 = 2.5 - 2.78388218141501*i
The graph