Mister Exam
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How to use it?
- Equation:
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Equation sinx=1
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Equation cosx=1/2
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Equation sqrt(x)+1=x-1
- Equation x^(2)-4*x+y^(2)+3=0
- Express {x} in terms of y where:
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Identical expressions
- 9x^ two +2x+ seven = zero
- 9x squared plus 2x plus 7 equally 0
- 9x to the power of two plus 2x plus seven equally zero
- 9x2+2x+7=0
- 9x²+2x+7=0
- 9x to the power of 2+2x+7=0
- 9x^2+2x+7=O
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Similar expressions
- 9x^2-2x+7=0
- 9x^2+2x-7=0
9x^2+2x+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
$$a = 9$$
$$b = 2$$
$$c = 7$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{1}{9} + \frac{\sqrt{62} i}{9}$$
$$x_{2} = - \frac{1}{9} - \frac{\sqrt{62} i}{9}$$
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
$$a = 9$$
$$b = 2$$
$$c = 7$$
, then
D = b^2 - 4 * a * c =
(2)^2 - 4 * (9) * (7) = -248
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
$$x_{1} = - \frac{1}{9} + \frac{\sqrt{62} i}{9}$$
$$x_{2} = - \frac{1}{9} - \frac{\sqrt{62} i}{9}$$
Vieta's Theorem
rewrite the equation
$$\left(9 x^{2} + 2 x\right) + 7 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{2 x}{9} + \frac{7}{9} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{2}{9}$$
$$q = \frac{c}{a}$$
$$q = \frac{7}{9}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{2}{9}$$
$$x_{1} x_{2} = \frac{7}{9}$$
$$\left(9 x^{2} + 2 x\right) + 7 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{2 x}{9} + \frac{7}{9} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{2}{9}$$
$$q = \frac{c}{a}$$
$$q = \frac{7}{9}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{2}{9}$$
$$x_{1} x_{2} = \frac{7}{9}$$
Rapid solution
[src]
____
1 I*\/ 62
x1 = - - - --------
9 9
$$x_{1} = - \frac{1}{9} - \frac{\sqrt{62} i}{9}$$
____
1 I*\/ 62
x2 = - - + --------
9 9
$$x_{2} = - \frac{1}{9} + \frac{\sqrt{62} i}{9}$$
x2 = -1/9 + sqrt(62)*i/9
Sum and product of roots
[src]
sum
____ ____ 1 I*\/ 62 1 I*\/ 62 - - - -------- + - - + -------- 9 9 9 9
$$\left(- \frac{1}{9} - \frac{\sqrt{62} i}{9}\right) + \left(- \frac{1}{9} + \frac{\sqrt{62} i}{9}\right)$$
=
-2/9
$$- \frac{2}{9}$$
product
/ ____\ / ____\ | 1 I*\/ 62 | | 1 I*\/ 62 | |- - - --------|*|- - + --------| \ 9 9 / \ 9 9 /
$$\left(- \frac{1}{9} - \frac{\sqrt{62} i}{9}\right) \left(- \frac{1}{9} + \frac{\sqrt{62} i}{9}\right)$$
=
7/9
$$\frac{7}{9}$$
7/9
Numerical answer
[src]
x1 = -0.111111111111111 + 0.87488976377909*i
x2 = -0.111111111111111 - 0.87488976377909*i
x2 = -0.111111111111111 - 0.87488976377909*i