Mister Exam
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Similar expressions
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1*x^2-10*x-90=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
$$a = 1$$
$$b = -10$$
$$c = -90$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 5 + \sqrt{115}$$
$$x_{2} = 5 - \sqrt{115}$$
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
$$a = 1$$
$$b = -10$$
$$c = -90$$
, then
D = b^2 - 4 * a * c =
(-10)^2 - 4 * (1) * (-90) = 460
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 5 + \sqrt{115}$$
$$x_{2} = 5 - \sqrt{115}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -10$$
$$q = \frac{c}{a}$$
$$q = -90$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 10$$
$$x_{1} x_{2} = -90$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -10$$
$$q = \frac{c}{a}$$
$$q = -90$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 10$$
$$x_{1} x_{2} = -90$$
Sum and product of roots
[src]
sum
_____ _____ 5 - \/ 115 + 5 + \/ 115
$$\left(5 - \sqrt{115}\right) + \left(5 + \sqrt{115}\right)$$
=
10
$$10$$
product
/ _____\ / _____\ \5 - \/ 115 /*\5 + \/ 115 /
$$\left(5 - \sqrt{115}\right) \left(5 + \sqrt{115}\right)$$
=
-90
$$-90$$
-90
Rapid solution
[src]
_____ x1 = 5 - \/ 115
$$x_{1} = 5 - \sqrt{115}$$
_____ x2 = 5 + \/ 115
$$x_{2} = 5 + \sqrt{115}$$
x2 = 5 + sqrt(115)