Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 5^x=8^x
-
Equation -27x+220=-5x
-
Equation x+12=5
-
Equation 3/(x-5)+8/x=2
- Express {x} in terms of y where:
- 12*x+1*y=15
- -16*x-10*y=4
- -4*x-20*y=10
- 19*x+6*y=8
- Factor polynomial:
- x^2-5*x-5
- Factor squared:
- x^2-5*x-5
-
Identical expressions
- x^ two - five *x- five = zero
- x squared minus 5 multiply by x minus 5 equally 0
- x to the power of two minus five multiply by x minus five equally zero
- x2-5*x-5=0
- x²-5*x-5=0
- x to the power of 2-5*x-5=0
- x^2-5x-5=0
- x2-5x-5=0
- x^2-5*x-5=O
-
Similar expressions
- x^2+5*x-5=0
- x^2-5*x+5=0
x^2-5*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
$$a = 1$$
$$b = -5$$
$$c = -5$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{5}{2} + \frac{3 \sqrt{5}}{2}$$
$$x_{2} = \frac{5}{2} - \frac{3 \sqrt{5}}{2}$$
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
$$a = 1$$
$$b = -5$$
$$c = -5$$
, then
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (1) * (-5) = 45
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{5}{2} + \frac{3 \sqrt{5}}{2}$$
$$x_{2} = \frac{5}{2} - \frac{3 \sqrt{5}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = -5$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = -5$$
Sum and product of roots
[src]
sum
___ ___ 5 3*\/ 5 5 3*\/ 5 - - ------- + - + ------- 2 2 2 2
$$\left(\frac{5}{2} - \frac{3 \sqrt{5}}{2}\right) + \left(\frac{5}{2} + \frac{3 \sqrt{5}}{2}\right)$$
=
5
$$5$$
product
/ ___\ / ___\ |5 3*\/ 5 | |5 3*\/ 5 | |- - -------|*|- + -------| \2 2 / \2 2 /
$$\left(\frac{5}{2} - \frac{3 \sqrt{5}}{2}\right) \left(\frac{5}{2} + \frac{3 \sqrt{5}}{2}\right)$$
=
-5
$$-5$$
-5
Rapid solution
[src]
___
5 3*\/ 5
x1 = - - -------
2 2
$$x_{1} = \frac{5}{2} - \frac{3 \sqrt{5}}{2}$$
___
5 3*\/ 5
x2 = - + -------
2 2
$$x_{2} = \frac{5}{2} + \frac{3 \sqrt{5}}{2}$$
x2 = 5/2 + 3*sqrt(5)/2
The graph