Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 2x-3=10
-
Equation 2*sin(x)=0
-
Equation 4*x=-12
-
Equation x^4-x^2+1=0
- Express {x} in terms of y where:
- -12*x-6*y=-3
- -7*x-11*y=18
- -13*x+8*y=-18
- -13*x+3*y=-4
-
Identical expressions
- -x^ two +4x+ nineteen
- minus x squared plus 4x plus 19
- minus x to the power of two plus 4x plus nineteen
- -x2+4x+19
- -x²+4x+19
- -x to the power of 2+4x+19
-
Similar expressions
- -x^2-4x+19
- x^2+4x+19
- -x^2+4x-19
-x^2+4x+19 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
$$a = -1$$
$$b = 4$$
$$c = 19$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 2 - \sqrt{23}$$
$$x_{2} = 2 + \sqrt{23}$$
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
$$a = -1$$
$$b = 4$$
$$c = 19$$
, then
D = b^2 - 4 * a * c =
(4)^2 - 4 * (-1) * (19) = 92
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 2 - \sqrt{23}$$
$$x_{2} = 2 + \sqrt{23}$$
Vieta's Theorem
rewrite the equation
$$\left(- x^{2} + 4 x\right) + 19 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x - 19 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -19$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = -19$$
$$\left(- x^{2} + 4 x\right) + 19 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x - 19 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -19$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = -19$$
Rapid solution
[src]
____ x1 = 2 - \/ 23
$$x_{1} = 2 - \sqrt{23}$$
____ x2 = 2 + \/ 23
$$x_{2} = 2 + \sqrt{23}$$
x2 = 2 + sqrt(23)
Sum and product of roots
[src]
sum
____ ____ 2 - \/ 23 + 2 + \/ 23
$$\left(2 - \sqrt{23}\right) + \left(2 + \sqrt{23}\right)$$
=
4
$$4$$
product
/ ____\ / ____\ \2 - \/ 23 /*\2 + \/ 23 /
$$\left(2 - \sqrt{23}\right) \left(2 + \sqrt{23}\right)$$
=
-19
$$-19$$
-19