Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation z^2-4*z+5=0
-
Equation (25+x)-10=40
-
Equation x^2+4*x-21=0
-
Equation 2^x=10
- Express {x} in terms of y where:
- 11*x+3*y=-16
- -2*x+13*y=-6
- 20*x-3*y=3
- 18*x-7*y=-4
- Graphing y =:
- x(x-5)
- 1-4x
-
Identical expressions
- x(x- five)= one -4x
- x(x minus 5) equally 1 minus 4x
- x(x minus five) equally one minus 4x
- xx-5=1-4x
-
Similar expressions
- 5(x-1)-4(x-2)=10
- x(x-5)=1+4x
- x*(x-5)=0
- x*x-5=2*x+19
- 6*x/(2*(1-1/(x+1)))-4*x/(2*(1+1/(x-1)))=97
- 3*(1-x)+5*(x+2)=1-4*x
- x(x+5)=1-4x
x(x-5)=1-4x equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$x \left(x - 5\right) = 1 - 4 x$$
to
$$x \left(x - 5\right) + \left(4 x - 1\right) = 0$$
Expand the expression in the equation
$$x \left(x - 5\right) + \left(4 x - 1\right) = 0$$
We get the quadratic equation
$$x^{2} - x - 1 = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{5}}{2}$$
left part with negative sign.
The equation is transformed from
$$x \left(x - 5\right) = 1 - 4 x$$
to
$$x \left(x - 5\right) + \left(4 x - 1\right) = 0$$
Expand the expression in the equation
$$x \left(x - 5\right) + \left(4 x - 1\right) = 0$$
We get the quadratic equation
$$x^{2} - x - 1 = 0$$
This equation is of the form
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
$$a = 1$$
$$b = -1$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (-1) = 5
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{5}}{2}$$
Sum and product of roots
[src]
sum
___ ___ 1 \/ 5 1 \/ 5 - - ----- + - + ----- 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)$$
=
1
$$1$$
product
/ ___\ / ___\ |1 \/ 5 | |1 \/ 5 | |- - -----|*|- + -----| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)$$
=
-1
$$-1$$
-1
Rapid solution
[src]
___
1 \/ 5
x1 = - - -----
2 2
$$x_{1} = \frac{1}{2} - \frac{\sqrt{5}}{2}$$
___
1 \/ 5
x2 = - + -----
2 2
$$x_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2}$$
x2 = 1/2 + sqrt(5)/2
The graph