Mister Exam
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How to use it?
- Equation:
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Identical expressions
- 8x^ two - sixty-four *x+ one thousand, nine hundred and twenty = zero
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Similar expressions
- 8x^2-64*x-1920=0
- 8x^2+64*x+1920=0
8x^2-64*x+1920=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
$$a = 8$$
$$b = -64$$
$$c = 1920$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = 4 + 4 \sqrt{14} i$$
$$x_{2} = 4 - 4 \sqrt{14} i$$
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
$$a = 8$$
$$b = -64$$
$$c = 1920$$
, then
D = b^2 - 4 * a * c =
(-64)^2 - 4 * (8) * (1920) = -57344
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
$$x_{1} = 4 + 4 \sqrt{14} i$$
$$x_{2} = 4 - 4 \sqrt{14} i$$
Vieta's Theorem
rewrite the equation
$$\left(8 x^{2} - 64 x\right) + 1920 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 8 x + 240 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -8$$
$$q = \frac{c}{a}$$
$$q = 240$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 8$$
$$x_{1} x_{2} = 240$$
$$\left(8 x^{2} - 64 x\right) + 1920 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 8 x + 240 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -8$$
$$q = \frac{c}{a}$$
$$q = 240$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 8$$
$$x_{1} x_{2} = 240$$
Rapid solution
[src]
____ x1 = 4 - 4*I*\/ 14
$$x_{1} = 4 - 4 \sqrt{14} i$$
____ x2 = 4 + 4*I*\/ 14
$$x_{2} = 4 + 4 \sqrt{14} i$$
x2 = 4 + 4*sqrt(14)*i
Sum and product of roots
[src]
sum
____ ____ 4 - 4*I*\/ 14 + 4 + 4*I*\/ 14
$$\left(4 - 4 \sqrt{14} i\right) + \left(4 + 4 \sqrt{14} i\right)$$
=
8
$$8$$
product
/ ____\ / ____\ \4 - 4*I*\/ 14 /*\4 + 4*I*\/ 14 /
$$\left(4 - 4 \sqrt{14} i\right) \left(4 + 4 \sqrt{14} i\right)$$
=
240
$$240$$
240
Numerical answer
[src]
x1 = 4.0 + 14.9666295470958*i
x2 = 4.0 - 14.9666295470958*i
x2 = 4.0 - 14.9666295470958*i