Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 9^x-8*3^x-9=0
- Equation x+y=4
-
Equation 5x+8=-12
-
Equation x/9+x=-3
- Express {x} in terms of y where:
- 18*x+17*y=-14
- 16*x-17*y=-15
- -4*x-4*y=19
- 15*x+1*y=19
-
Identical expressions
- x^ two + six *x- nineteen = zero
- x squared plus 6 multiply by x minus 19 equally 0
- x to the power of two plus six multiply by x minus nineteen equally zero
- x2+6*x-19=0
- x²+6*x-19=0
- x to the power of 2+6*x-19=0
- x^2+6x-19=0
- x2+6x-19=0
- x^2+6*x-19=O
-
Similar expressions
- x^2+6*x+19=0
- x^2-6*x-19=0
x^2+6*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
$$a = 1$$
$$b = 6$$
$$c = -19$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = -3 + 2 \sqrt{7}$$
$$x_{2} = - 2 \sqrt{7} - 3$$
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
$$a = 1$$
$$b = 6$$
$$c = -19$$
, then
D = b^2 - 4 * a * c =
(6)^2 - 4 * (1) * (-19) = 112
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = -3 + 2 \sqrt{7}$$
$$x_{2} = - 2 \sqrt{7} - 3$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 6$$
$$q = \frac{c}{a}$$
$$q = -19$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -6$$
$$x_{1} x_{2} = -19$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 6$$
$$q = \frac{c}{a}$$
$$q = -19$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -6$$
$$x_{1} x_{2} = -19$$
Sum and product of roots
[src]
sum
___ ___ -3 + 2*\/ 7 + -3 - 2*\/ 7
$$\left(- 2 \sqrt{7} - 3\right) + \left(-3 + 2 \sqrt{7}\right)$$
=
-6
$$-6$$
product
/ ___\ / ___\ \-3 + 2*\/ 7 /*\-3 - 2*\/ 7 /
$$\left(-3 + 2 \sqrt{7}\right) \left(- 2 \sqrt{7} - 3\right)$$
=
-19
$$-19$$
-19
Rapid solution
[src]
___ x1 = -3 + 2*\/ 7
$$x_{1} = -3 + 2 \sqrt{7}$$
___ x2 = -3 - 2*\/ 7
$$x_{2} = - 2 \sqrt{7} - 3$$
x2 = -2*sqrt(7) - 3