Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 2x-3(x+3)=-5
- Equation x+y=5
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Equation 1*(x-1)*(x-3)=0
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Equation 5x+12=8x+30
- Express {x} in terms of y where:
- -17*x+2*y=-4
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Identical expressions
- 5x^ two +3x+ six = zero
- 5x squared plus 3x plus 6 equally 0
- 5x to the power of two plus 3x plus six equally zero
- 5x2+3x+6=0
- 5x²+3x+6=0
- 5x to the power of 2+3x+6=0
- 5x^2+3x+6=O
-
Similar expressions
- 5x^2+3x-6=0
- 5x^2-3x+6=0
5x^2+3x+6=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
$$a = 5$$
$$b = 3$$
$$c = 6$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{3}{10} + \frac{\sqrt{111} i}{10}$$
Simplify
$$x_{2} = - \frac{3}{10} - \frac{\sqrt{111} i}{10}$$
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
$$a = 5$$
$$b = 3$$
$$c = 6$$
, then
D = b^2 - 4 * a * c =
(3)^2 - 4 * (5) * (6) = -111
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
$$x_{1} = - \frac{3}{10} + \frac{\sqrt{111} i}{10}$$
Simplify
$$x_{2} = - \frac{3}{10} - \frac{\sqrt{111} i}{10}$$
Simplify
Vieta's Theorem
rewrite the equation
$$5 x^{2} + 3 x + 6 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{3 x}{5} + \frac{6}{5} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{3}{5}$$
$$q = \frac{c}{a}$$
$$q = \frac{6}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{3}{5}$$
$$x_{1} x_{2} = \frac{6}{5}$$
$$5 x^{2} + 3 x + 6 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{3 x}{5} + \frac{6}{5} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{3}{5}$$
$$q = \frac{c}{a}$$
$$q = \frac{6}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{3}{5}$$
$$x_{1} x_{2} = \frac{6}{5}$$
Rapid solution
[src]
_____
3 I*\/ 111
x1 = - -- - ---------
10 10
$$x_{1} = - \frac{3}{10} - \frac{\sqrt{111} i}{10}$$
_____
3 I*\/ 111
x2 = - -- + ---------
10 10
$$x_{2} = - \frac{3}{10} + \frac{\sqrt{111} i}{10}$$
Sum and product of roots
[src]
sum
_____ _____
3 I*\/ 111 3 I*\/ 111
0 + - -- - --------- + - -- + ---------
10 10 10 10
$$\left(0 - \left(\frac{3}{10} + \frac{\sqrt{111} i}{10}\right)\right) - \left(\frac{3}{10} - \frac{\sqrt{111} i}{10}\right)$$
=
-3/5
$$- \frac{3}{5}$$
product
/ _____\ / _____\ | 3 I*\/ 111 | | 3 I*\/ 111 | 1*|- -- - ---------|*|- -- + ---------| \ 10 10 / \ 10 10 /
$$1 \left(- \frac{3}{10} - \frac{\sqrt{111} i}{10}\right) \left(- \frac{3}{10} + \frac{\sqrt{111} i}{10}\right)$$
=
6/5
$$\frac{6}{5}$$
6/5
Numerical answer
[src]
x1 = -0.3 - 1.05356537528527*i
x2 = -0.3 + 1.05356537528527*i
x2 = -0.3 + 1.05356537528527*i
The graph