Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^3+4*x^2-x-4=0
-
Equation (x-7)^2=(9-x)^2
-
Equation 3*x+50=77
-
Equation 5-2(x-1)=4-x
- Express {x} in terms of y where:
- 18*x-12*y=-20
- -19*x+1*y=0
- -12*x-3*y=-9
- 18*x-20*y=-7
- Factor polynomial:
- y^2-81
-
Identical expressions
- y^ two - eighty-one
- y squared minus 81
- y to the power of two minus eighty minus one
- y2-81
- y²-81
- y to the power of 2-81
-
Similar expressions
- y^2+81
y^2-81 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:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 0$$
$$c = -81$$
, then
Because D > 0, then the equation has two roots.
or
$$y_{1} = 9$$
$$y_{2} = -9$$
a*y^2 + b*y + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 0$$
$$c = -81$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (-81) = 324
Because D > 0, then the equation has two roots.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
or
$$y_{1} = 9$$
$$y_{2} = -9$$
Vieta's Theorem
it is reduced quadratic equation
$$p y + q + y^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -81$$
Vieta Formulas
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = 0$$
$$y_{1} y_{2} = -81$$
$$p y + q + y^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -81$$
Vieta Formulas
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = 0$$
$$y_{1} y_{2} = -81$$
Sum and product of roots
[src]
sum
-9 + 9
$$-9 + 9$$
=
0
$$0$$
product
-9*9
$$- 81$$
=
-81
$$-81$$
-81