Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 16x-3=8x-43
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Equation (|x+4|)=5
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Equation -2*x-4=3*x
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Equation x^2+4*x+3=0
- Express {x} in terms of y where:
- -10*x+13*y=5
- -10*x-2*y=18
- 20*x-16*y=7
- 1*x+11*y=-3
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Identical expressions
- k^ two + two *k+ five = zero
- k squared plus 2 multiply by k plus 5 equally 0
- k to the power of two plus two multiply by k plus five equally zero
- k2+2*k+5=0
- k²+2*k+5=0
- k to the power of 2+2*k+5=0
- k^2+2k+5=0
- k2+2k+5=0
- k^2+2*k+5=O
-
Similar expressions
- k^2-2*k+5=0
- k^2+2*k-5=0
k^2+2*k+5=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
This equation is of the form
$$a\ k^2 + b\ k + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$k_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$k_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = 2$$
$$c = 5$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right) 1 \cdot 4 \cdot 5 + 2^{2} = -16$$
Because D<0, then the equation
has no real roots,
but complex roots is exists.
$$k_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$k_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$k_{1} = -1 + 2 i$$
Simplify
$$k_{2} = -1 - 2 i$$
Simplify
$$a\ k^2 + b\ k + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$k_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$k_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = 2$$
$$c = 5$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right) 1 \cdot 4 \cdot 5 + 2^{2} = -16$$
Because D<0, then the equation
has no real roots,
but complex roots is exists.
$$k_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$k_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$k_{1} = -1 + 2 i$$
Simplify
$$k_{2} = -1 - 2 i$$
Simplify
Vieta's Theorem
it is reduced quadratic equation
$$k^{2} + k p + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = 5$$
Vieta Formulas
$$k_{1} + k_{2} = - p$$
$$k_{1} k_{2} = q$$
$$k_{1} + k_{2} = -2$$
$$k_{1} k_{2} = 5$$
$$k^{2} + k p + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = 5$$
Vieta Formulas
$$k_{1} + k_{2} = - p$$
$$k_{1} k_{2} = q$$
$$k_{1} + k_{2} = -2$$
$$k_{1} k_{2} = 5$$
Sum and product of roots
[src]
sum
-1 - 2*I + -1 + 2*I
$$\left(-1 - 2 i\right) + \left(-1 + 2 i\right)$$
=
-2
$$-2$$
product
-1 - 2*I * -1 + 2*I
$$\left(-1 - 2 i\right) * \left(-1 + 2 i\right)$$
=
5
$$5$$
The graph