Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation (a-2)(b+5)=(a-2)b+(a-2)5
-
Equation 5=4x-3x
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Equation x^3+6*x^2=4*x+24
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Equation 11/(x+4)=-11/7
- Express {x} in terms of y where:
- -5*x+3*y=10
- 17*x+11*y=12
- 14*x+5*y=-16
- -7*x+6*y=13
-
Identical expressions
- x^ two - five *x- nineteen = zero
- x squared minus 5 multiply by x minus 19 equally 0
- x to the power of two minus five multiply by x minus nineteen equally zero
- x2-5*x-19=0
- x²-5*x-19=0
- x to the power of 2-5*x-19=0
- x^2-5x-19=0
- x2-5x-19=0
- x^2-5*x-19=O
-
Similar expressions
- x^2+5*x-19=0
- x^2-5*x+19=0
x^2-5*x-19=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
$$a = 1$$
$$b = -5$$
$$c = -19$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{101}}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{101}}{2}$$
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
$$a = 1$$
$$b = -5$$
$$c = -19$$
, then
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (1) * (-19) = 101
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{101}}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{101}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -19$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = -19$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -19$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = -19$$
Sum and product of roots
[src]
sum
_____ _____ 5 \/ 101 5 \/ 101 - - ------- + - + ------- 2 2 2 2
$$\left(\frac{5}{2} - \frac{\sqrt{101}}{2}\right) + \left(\frac{5}{2} + \frac{\sqrt{101}}{2}\right)$$
=
5
$$5$$
product
/ _____\ / _____\ |5 \/ 101 | |5 \/ 101 | |- - -------|*|- + -------| \2 2 / \2 2 /
$$\left(\frac{5}{2} - \frac{\sqrt{101}}{2}\right) \left(\frac{5}{2} + \frac{\sqrt{101}}{2}\right)$$
=
-19
$$-19$$
-19
Rapid solution
[src]
_____
5 \/ 101
x1 = - - -------
2 2
$$x_{1} = \frac{5}{2} - \frac{\sqrt{101}}{2}$$
_____
5 \/ 101
x2 = - + -------
2 2
$$x_{2} = \frac{5}{2} + \frac{\sqrt{101}}{2}$$
x2 = 5/2 + sqrt(101)/2