Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x+19=12
- Equation x-y=1
-
Equation 2-3(7+2x)=11
-
Equation tg(x)=1
- Express {x} in terms of y where:
- 17*x-20*y=6
- 18*x-8*y=-2
- -11*x+15*y=7
- 4*x+2*y=5
-
Identical expressions
- - twelve *x^ two - ten *x- nine = zero
- minus 12 multiply by x squared minus 10 multiply by x minus 9 equally 0
- minus twelve multiply by x to the power of two minus ten multiply by x minus nine equally zero
- -12*x2-10*x-9=0
- -12*x²-10*x-9=0
- -12*x to the power of 2-10*x-9=0
- -12x^2-10x-9=0
- -12x2-10x-9=0
- -12*x^2-10*x-9=O
-
Similar expressions
- -12*x^2-10*x+9=0
- -12*x^2+10*x-9=0
- 12*x^2-10*x-9=0
-12*x^2-10*x-9=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
$$a = -12$$
$$b = -10$$
$$c = -9$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{5}{12} - \frac{\sqrt{83} i}{12}$$
$$x_{2} = - \frac{5}{12} + \frac{\sqrt{83} i}{12}$$
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
$$a = -12$$
$$b = -10$$
$$c = -9$$
, then
D = b^2 - 4 * a * c =
(-10)^2 - 4 * (-12) * (-9) = -332
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{5}{12} - \frac{\sqrt{83} i}{12}$$
$$x_{2} = - \frac{5}{12} + \frac{\sqrt{83} i}{12}$$
Vieta's Theorem
rewrite the equation
$$\left(- 12 x^{2} - 10 x\right) - 9 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{5 x}{6} + \frac{3}{4} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{5}{6}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{4}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{5}{6}$$
$$x_{1} x_{2} = \frac{3}{4}$$
$$\left(- 12 x^{2} - 10 x\right) - 9 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{5 x}{6} + \frac{3}{4} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{5}{6}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{4}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{5}{6}$$
$$x_{1} x_{2} = \frac{3}{4}$$
Rapid solution
[src]
____
5 I*\/ 83
x1 = - -- - --------
12 12
$$x_{1} = - \frac{5}{12} - \frac{\sqrt{83} i}{12}$$
____
5 I*\/ 83
x2 = - -- + --------
12 12
$$x_{2} = - \frac{5}{12} + \frac{\sqrt{83} i}{12}$$
x2 = -5/12 + sqrt(83)*i/12
Sum and product of roots
[src]
sum
____ ____ 5 I*\/ 83 5 I*\/ 83 - -- - -------- + - -- + -------- 12 12 12 12
$$\left(- \frac{5}{12} - \frac{\sqrt{83} i}{12}\right) + \left(- \frac{5}{12} + \frac{\sqrt{83} i}{12}\right)$$
=
-5/6
$$- \frac{5}{6}$$
product
/ ____\ / ____\ | 5 I*\/ 83 | | 5 I*\/ 83 | |- -- - --------|*|- -- + --------| \ 12 12 / \ 12 12 /
$$\left(- \frac{5}{12} - \frac{\sqrt{83} i}{12}\right) \left(- \frac{5}{12} + \frac{\sqrt{83} i}{12}\right)$$
=
3/4
$$\frac{3}{4}$$
3/4
Numerical answer
[src]
x1 = -0.416666666666667 - 0.759202798262025*i
x2 = -0.416666666666667 + 0.759202798262025*i
x2 = -0.416666666666667 + 0.759202798262025*i