Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 3*x^2-x-5=0
-
Equation -x^3+14*x^2-61*x+84=0
-
Equation 2x^2+6x=0
-
Equation x^2-14x+40=0
- Express {x} in terms of y where:
- 13*x-2*y=5
- 5*x+4*y=9
- 8*x+13*y=-6
- 17*x-5*y=11
-
Identical expressions
- four . two x²+ four .9x+2. one = zero
- 4.2x² plus 4.9x plus 2.1 equally 0
- four . two x² plus four .9x plus 2. one equally zero
- 4.2x²+4.9x+2.1=O
-
Similar expressions
- 4.2x²-4.9x+2.1=0
- 4.2x²+4.9x-2.1=0
4.2x²+4.9x+2.1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 21*x 49*x 21 ----- + ---- + -- = 0 5 10 10
$$\left(\frac{21 x^{2}}{5} + \frac{49 x}{10}\right) + \frac{21}{10} = 0$$
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
$$a = \frac{21}{5}$$
$$b = \frac{49}{10}$$
$$c = \frac{21}{10}$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{7}{12} + \frac{\sqrt{23} i}{12}$$
$$x_{2} = - \frac{7}{12} - \frac{\sqrt{23} i}{12}$$
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
$$a = \frac{21}{5}$$
$$b = \frac{49}{10}$$
$$c = \frac{21}{10}$$
, then
D = b^2 - 4 * a * c =
(49/10)^2 - 4 * (21/5) * (21/10) = -1127/100
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
$$x_{1} = - \frac{7}{12} + \frac{\sqrt{23} i}{12}$$
$$x_{2} = - \frac{7}{12} - \frac{\sqrt{23} i}{12}$$
Vieta's Theorem
rewrite the equation
$$\left(\frac{21 x^{2}}{5} + \frac{49 x}{10}\right) + \frac{21}{10} = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{7 x}{6} + \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{7}{6}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{7}{6}$$
$$x_{1} x_{2} = \frac{1}{2}$$
$$\left(\frac{21 x^{2}}{5} + \frac{49 x}{10}\right) + \frac{21}{10} = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{7 x}{6} + \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{7}{6}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{7}{6}$$
$$x_{1} x_{2} = \frac{1}{2}$$
Sum and product of roots
[src]
sum
____ ____ 7 I*\/ 23 7 I*\/ 23 - -- - -------- + - -- + -------- 12 12 12 12
$$\left(- \frac{7}{12} - \frac{\sqrt{23} i}{12}\right) + \left(- \frac{7}{12} + \frac{\sqrt{23} i}{12}\right)$$
=
-7/6
$$- \frac{7}{6}$$
product
/ ____\ / ____\ | 7 I*\/ 23 | | 7 I*\/ 23 | |- -- - --------|*|- -- + --------| \ 12 12 / \ 12 12 /
$$\left(- \frac{7}{12} - \frac{\sqrt{23} i}{12}\right) \left(- \frac{7}{12} + \frac{\sqrt{23} i}{12}\right)$$
=
1/2
$$\frac{1}{2}$$
1/2
Rapid solution
[src]
____
7 I*\/ 23
x1 = - -- - --------
12 12
$$x_{1} = - \frac{7}{12} - \frac{\sqrt{23} i}{12}$$
____
7 I*\/ 23
x2 = - -- + --------
12 12
$$x_{2} = - \frac{7}{12} + \frac{\sqrt{23} i}{12}$$
x2 = -7/12 + sqrt(23)*i/12
Numerical answer
[src]
x1 = -0.583333333333333 - 0.399652626942727*i
x2 = -0.583333333333333 + 0.399652626942727*i
x2 = -0.583333333333333 + 0.399652626942727*i