Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 1-5x=-6x+8
-
Equation 12-x^2=11
-
Equation x^2-8x=0
-
Equation x^2=-3
- Express {x} in terms of y where:
- 16*x-7*y=-5
- -10*x+9*y=-6
- 1*x+4*y=20
- -2*x+1*y=-8
-
Identical expressions
- 3x^ two + seven = zero
- 3x squared plus 7 equally 0
- 3x to the power of two plus seven equally zero
- 3x2+7=0
- 3x²+7=0
- 3x to the power of 2+7=0
- 3x^2+7=O
-
Similar expressions
- 3x^2-7=0
3x^2+7=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
$$a = 3$$
$$b = 0$$
$$c = 7$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{\sqrt{21} i}{3}$$
$$x_{2} = - \frac{\sqrt{21} i}{3}$$
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
$$a = 3$$
$$b = 0$$
$$c = 7$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (3) * (7) = -84
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
$$x_{1} = \frac{\sqrt{21} i}{3}$$
$$x_{2} = - \frac{\sqrt{21} i}{3}$$
Vieta's Theorem
rewrite the equation
$$3 x^{2} + 7 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{7}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{7}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = \frac{7}{3}$$
$$3 x^{2} + 7 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{7}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{7}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = \frac{7}{3}$$
Sum and product of roots
[src]
sum
____ ____
I*\/ 21 I*\/ 21
- -------- + --------
3 3
$$- \frac{\sqrt{21} i}{3} + \frac{\sqrt{21} i}{3}$$
=
0
$$0$$
product
____ ____
-I*\/ 21 I*\/ 21
----------*--------
3 3
$$- \frac{\sqrt{21} i}{3} \frac{\sqrt{21} i}{3}$$
=
7/3
$$\frac{7}{3}$$
7/3
Rapid solution
[src]
____
-I*\/ 21
x1 = ----------
3
$$x_{1} = - \frac{\sqrt{21} i}{3}$$
____
I*\/ 21
x2 = --------
3
$$x_{2} = \frac{\sqrt{21} i}{3}$$
x2 = sqrt(21)*i/3
The graph