Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 1/(x-3)^2-3/(x-3)-4=0
-
Equation x^4+2*x^2-8=0
-
Equation (x-2)^5+(4-x)^5=2
-
Equation 6/(x+5)=-5
- Express {x} in terms of y where:
- 17*x-20*y=6
- 5*x-13*y=17
- 6*x+17*y=3
- 1*x+4*y=20
- Factor polynomial:
- x^2-x-4
- Factor squared:
- x^2-x-4
-
Identical expressions
- x^ two -x- four = zero
- x squared minus x minus 4 equally 0
- x to the power of two minus x minus four equally zero
- x2-x-4=0
- x²-x-4=0
- x to the power of 2-x-4=0
- x^2-x-4=O
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Similar expressions
- x^2+x-4=0
- x^2-x+4=0
x^2-x-4=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
$$a = 1$$
$$b = -1$$
$$c = -4$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{17}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{17}}{2}$$
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
$$a = 1$$
$$b = -1$$
$$c = -4$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (-4) = 17
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{17}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{17}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -4$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = -4$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -4$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = -4$$
Rapid solution
[src]
____
1 \/ 17
x1 = - - ------
2 2
$$x_{1} = \frac{1}{2} - \frac{\sqrt{17}}{2}$$
____
1 \/ 17
x2 = - + ------
2 2
$$x_{2} = \frac{1}{2} + \frac{\sqrt{17}}{2}$$
x2 = 1/2 + sqrt(17)/2
Sum and product of roots
[src]
sum
____ ____ 1 \/ 17 1 \/ 17 - - ------ + - + ------ 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{17}}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{17}}{2}\right)$$
=
1
$$1$$
product
/ ____\ / ____\ |1 \/ 17 | |1 \/ 17 | |- - ------|*|- + ------| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{17}}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{17}}{2}\right)$$
=
-4
$$-4$$
-4
The graph