Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 4(x-1)=2(2x-8)+12
-
Equation x/4+x=12
-
Equation 5^x=6-x
-
Equation 3*x=0
- Express {x} in terms of y where:
- -16*x-9*y=-1
- 4*x+13*y=8
- 14*x+17*y=19
- 3*x-11*y=20
- Integral of d{x}:
-
x^2+3x+1
-
Identical expressions
- x^ two +3x+ one
- x squared plus 3x plus 1
- x to the power of two plus 3x plus one
- x2+3x+1
- x²+3x+1
- x to the power of 2+3x+1
-
Similar expressions
- x^2-3x+1
- x^2+3x-1
x^2+3x+1 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
$$a = 1$$
$$b = 3$$
$$c = 1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{3}{2} + \frac{\sqrt{5}}{2}$$
$$x_{2} = - \frac{3}{2} - \frac{\sqrt{5}}{2}$$
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
$$a = 1$$
$$b = 3$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(3)^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
$$x_{1} = - \frac{3}{2} + \frac{\sqrt{5}}{2}$$
$$x_{2} = - \frac{3}{2} - \frac{\sqrt{5}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3$$
$$x_{1} x_{2} = 1$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -3$$
$$x_{1} x_{2} = 1$$
Rapid solution
[src]
___
3 \/ 5
x1 = - - - -----
2 2
$$x_{1} = - \frac{3}{2} - \frac{\sqrt{5}}{2}$$
___
3 \/ 5
x2 = - - + -----
2 2
$$x_{2} = - \frac{3}{2} + \frac{\sqrt{5}}{2}$$
x2 = -3/2 + sqrt(5)/2
Sum and product of roots
[src]
sum
___ ___ 3 \/ 5 3 \/ 5 - - - ----- + - - + ----- 2 2 2 2
$$\left(- \frac{3}{2} - \frac{\sqrt{5}}{2}\right) + \left(- \frac{3}{2} + \frac{\sqrt{5}}{2}\right)$$
=
-3
$$-3$$
product
/ ___\ / ___\ | 3 \/ 5 | | 3 \/ 5 | |- - - -----|*|- - + -----| \ 2 2 / \ 2 2 /
$$\left(- \frac{3}{2} - \frac{\sqrt{5}}{2}\right) \left(- \frac{3}{2} + \frac{\sqrt{5}}{2}\right)$$
=
1
$$1$$
1