Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x^2+2x+1=0
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Equation sinx=1
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Equation -2*(-2-y)-y-2=0
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Equation cosx=1/2
- Express {x} in terms of y where:
- -7*x-8*y=19
- -6*x-18*y=-1
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- -5*x-11*y=-11
- Graphing y =:
- 2x^2-x+3
-
Identical expressions
- two x^2-x+ three = zero
- 2x squared minus x plus 3 equally 0
- two x squared minus x plus three equally zero
- 2x2-x+3=0
- 2x²-x+3=0
- 2x to the power of 2-x+3=0
- 2x^2-x+3=O
-
Similar expressions
- 2x^2+x+3=0
- 2x^2-x-3=0
2x^2-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
$$a = 2$$
$$b = -1$$
$$c = 3$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{4} + \frac{\sqrt{23} i}{4}$$
$$x_{2} = \frac{1}{4} - \frac{\sqrt{23} i}{4}$$
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
$$a = 2$$
$$b = -1$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (2) * (3) = -23
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
$$x_{1} = \frac{1}{4} + \frac{\sqrt{23} i}{4}$$
$$x_{2} = \frac{1}{4} - \frac{\sqrt{23} i}{4}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x^{2} - x\right) + 3 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{2} + \frac{3}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{2}$$
$$x_{1} x_{2} = \frac{3}{2}$$
$$\left(2 x^{2} - x\right) + 3 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{2} + \frac{3}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{2}$$
$$x_{1} x_{2} = \frac{3}{2}$$
Rapid solution
[src]
____
1 I*\/ 23
x1 = - - --------
4 4
$$x_{1} = \frac{1}{4} - \frac{\sqrt{23} i}{4}$$
____
1 I*\/ 23
x2 = - + --------
4 4
$$x_{2} = \frac{1}{4} + \frac{\sqrt{23} i}{4}$$
x2 = 1/4 + sqrt(23)*i/4
Sum and product of roots
[src]
sum
____ ____ 1 I*\/ 23 1 I*\/ 23 - - -------- + - + -------- 4 4 4 4
$$\left(\frac{1}{4} - \frac{\sqrt{23} i}{4}\right) + \left(\frac{1}{4} + \frac{\sqrt{23} i}{4}\right)$$
=
1/2
$$\frac{1}{2}$$
product
/ ____\ / ____\ |1 I*\/ 23 | |1 I*\/ 23 | |- - --------|*|- + --------| \4 4 / \4 4 /
$$\left(\frac{1}{4} - \frac{\sqrt{23} i}{4}\right) \left(\frac{1}{4} + \frac{\sqrt{23} i}{4}\right)$$
=
3/2
$$\frac{3}{2}$$
3/2
Numerical answer
[src]
x1 = 0.25 - 1.19895788082818*i
x2 = 0.25 + 1.19895788082818*i
x2 = 0.25 + 1.19895788082818*i
The graph