Mister Exam
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How to use it?
- Equation:
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Equation (x^2+3)/(x^2+1)=2
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Equation 4*x-8=x+1
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Equation -4x+9=-19
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Identical expressions
- -10x-3x^ two = seventy
- minus 10x minus 3x squared equally 70
- minus 10x minus 3x to the power of two equally seventy
- -10x-3x2=70
- -10x-3x²=70
- -10x-3x to the power of 2=70
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Similar expressions
- 10x-3x^2=70
- -10x+3x^2=70
-10x-3x^2=70 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$- 3 x^{2} - 10 x = 70$$
to
$$\left(- 3 x^{2} - 10 x\right) - 70 = 0$$
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
$$a = -3$$
$$b = -10$$
$$c = -70$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{5}{3} - \frac{\sqrt{185} i}{3}$$
$$x_{2} = - \frac{5}{3} + \frac{\sqrt{185} i}{3}$$
left part with negative sign.
The equation is transformed from
$$- 3 x^{2} - 10 x = 70$$
to
$$\left(- 3 x^{2} - 10 x\right) - 70 = 0$$
This equation is of the form
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
$$a = -3$$
$$b = -10$$
$$c = -70$$
, then
D = b^2 - 4 * a * c =
(-10)^2 - 4 * (-3) * (-70) = -740
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
$$x_{1} = - \frac{5}{3} - \frac{\sqrt{185} i}{3}$$
$$x_{2} = - \frac{5}{3} + \frac{\sqrt{185} i}{3}$$
Vieta's Theorem
rewrite the equation
$$- 3 x^{2} - 10 x = 70$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{10 x}{3} + \frac{70}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{10}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{70}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{10}{3}$$
$$x_{1} x_{2} = \frac{70}{3}$$
$$- 3 x^{2} - 10 x = 70$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{10 x}{3} + \frac{70}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{10}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{70}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{10}{3}$$
$$x_{1} x_{2} = \frac{70}{3}$$
Rapid solution
[src]
_____
5 I*\/ 185
x1 = - - - ---------
3 3
$$x_{1} = - \frac{5}{3} - \frac{\sqrt{185} i}{3}$$
_____
5 I*\/ 185
x2 = - - + ---------
3 3
$$x_{2} = - \frac{5}{3} + \frac{\sqrt{185} i}{3}$$
x2 = -5/3 + sqrt(185)*i/3
Sum and product of roots
[src]
sum
_____ _____ 5 I*\/ 185 5 I*\/ 185 - - - --------- + - - + --------- 3 3 3 3
$$\left(- \frac{5}{3} - \frac{\sqrt{185} i}{3}\right) + \left(- \frac{5}{3} + \frac{\sqrt{185} i}{3}\right)$$
=
-10/3
$$- \frac{10}{3}$$
product
/ _____\ / _____\ | 5 I*\/ 185 | | 5 I*\/ 185 | |- - - ---------|*|- - + ---------| \ 3 3 / \ 3 3 /
$$\left(- \frac{5}{3} - \frac{\sqrt{185} i}{3}\right) \left(- \frac{5}{3} + \frac{\sqrt{185} i}{3}\right)$$
=
70/3
$$\frac{70}{3}$$
70/3
Numerical answer
[src]
x1 = -1.66666666666667 + 4.53382350291181*i
x2 = -1.66666666666667 - 4.53382350291181*i
x2 = -1.66666666666667 - 4.53382350291181*i