Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation tan(x)=-1/2
-
Equation 1*(x-1)*(x-3)=0
-
Equation 5x+12=8x+30
-
Equation x^2-11*x+30=0
- Express {x} in terms of y where:
- -2*x-19*y=3
- -2*x+18*y=13
- -2*x-15*y=-1
- 10*x-2*y=11
- Factor polynomial:
- 2*x^2-x-1
- Factor squared:
- 2*x^2-x-1
-
Identical expressions
- two *x^ two -x- one = zero
- 2 multiply by x squared minus x minus 1 equally 0
- two multiply by x to the power of two minus x minus one equally zero
- 2*x2-x-1=0
- 2*x²-x-1=0
- 2*x to the power of 2-x-1=0
- 2x^2-x-1=0
- 2x2-x-1=0
- 2*x^2-x-1=O
-
Similar expressions
- 2*x^2+x-1=0
- 2*x^2-x+1=0
2*x^2-x-1=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
$$a = 2$$
$$b = -1$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 1$$
$$x_{2} = - \frac{1}{2}$$
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
$$a = 2$$
$$b = -1$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (2) * (-1) = 9
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 1$$
$$x_{2} = - \frac{1}{2}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x^{2} - x\right) - 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{2} - \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{2}$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{2}$$
$$x_{1} x_{2} = - \frac{1}{2}$$
$$\left(2 x^{2} - x\right) - 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{x}{2} - \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{1}{2}$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{2}$$
$$x_{1} x_{2} = - \frac{1}{2}$$
Sum and product of roots
[src]
sum
1 - 1/2
$$- \frac{1}{2} + 1$$
=
1/2
$$\frac{1}{2}$$
product
-1/2
$$- \frac{1}{2}$$
=
-1/2
$$- \frac{1}{2}$$
-1/2
The graph