Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 6*x^2+x-7=0
-
Equation (x^2-9)^2+(x^2+x-6)^2=0
-
Equation (x-5)=(x+10)^2
-
Equation x^3+4*x^2+x-6=0
- Express {x} in terms of y where:
- 2*x+20*y=16
- -15*x-12*y=17
- -20*x+5*y=-11
- 2*x+10*y=-5
- Factor squared:
- x^2-x-5
- Factor polynomial:
- x^2-x-5
-
Identical expressions
- x^ two -x- five = zero
- x squared minus x minus 5 equally 0
- x to the power of two minus x minus five equally zero
- x2-x-5=0
- x²-x-5=0
- x to the power of 2-x-5=0
- x^2-x-5=O
-
Similar expressions
- x^2-x+5=0
- x^2+x-5=0
x^2-x-5=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
$$a = 1$$
$$b = -1$$
$$c = -5$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{21}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{21}}{2}$$
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
$$a = 1$$
$$b = -1$$
$$c = -5$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (-5) = 21
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{21}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{21}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = -5$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = -5$$
Sum and product of roots
[src]
sum
____ ____ 1 \/ 21 1 \/ 21 - - ------ + - + ------ 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{21}}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{21}}{2}\right)$$
=
1
$$1$$
product
/ ____\ / ____\ |1 \/ 21 | |1 \/ 21 | |- - ------|*|- + ------| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{21}}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{21}}{2}\right)$$
=
-5
$$-5$$
-5
Rapid solution
[src]
____
1 \/ 21
x1 = - - ------
2 2
$$x_{1} = \frac{1}{2} - \frac{\sqrt{21}}{2}$$
____
1 \/ 21
x2 = - + ------
2 2
$$x_{2} = \frac{1}{2} + \frac{\sqrt{21}}{2}$$
x2 = 1/2 + sqrt(21)/2
The graph