Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation (x+6)(2x+6)-20=0
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Equation 2x^2-9x+15=0
- Equation arctg(x/y)
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Equation sinx=1/3
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Identical expressions
- two x^2-9x+ fifteen = zero
- 2x squared minus 9x plus 15 equally 0
- two x squared minus 9x plus fifteen equally zero
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Similar expressions
- 2x^2-9x-15=0
- 2x^2+9x+15=0
2x^2-9x+15=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
$$a = 2$$
$$b = -9$$
$$c = 15$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{9}{4} + \frac{\sqrt{39} i}{4}$$
$$x_{2} = \frac{9}{4} - \frac{\sqrt{39} i}{4}$$
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
$$a = 2$$
$$b = -9$$
$$c = 15$$
, then
D = b^2 - 4 * a * c =
(-9)^2 - 4 * (2) * (15) = -39
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
$$x_{1} = \frac{9}{4} + \frac{\sqrt{39} i}{4}$$
$$x_{2} = \frac{9}{4} - \frac{\sqrt{39} i}{4}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x^{2} - 9 x\right) + 15 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{9 x}{2} + \frac{15}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{9}{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{15}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{9}{2}$$
$$x_{1} x_{2} = \frac{15}{2}$$
$$\left(2 x^{2} - 9 x\right) + 15 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{9 x}{2} + \frac{15}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{9}{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{15}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{9}{2}$$
$$x_{1} x_{2} = \frac{15}{2}$$
Sum and product of roots
[src]
sum
____ ____ 9 I*\/ 39 9 I*\/ 39 - - -------- + - + -------- 4 4 4 4
$$\left(\frac{9}{4} - \frac{\sqrt{39} i}{4}\right) + \left(\frac{9}{4} + \frac{\sqrt{39} i}{4}\right)$$
=
9/2
$$\frac{9}{2}$$
product
/ ____\ / ____\ |9 I*\/ 39 | |9 I*\/ 39 | |- - --------|*|- + --------| \4 4 / \4 4 /
$$\left(\frac{9}{4} - \frac{\sqrt{39} i}{4}\right) \left(\frac{9}{4} + \frac{\sqrt{39} i}{4}\right)$$
=
15/2
$$\frac{15}{2}$$
15/2
Rapid solution
[src]
____
9 I*\/ 39
x1 = - - --------
4 4
$$x_{1} = \frac{9}{4} - \frac{\sqrt{39} i}{4}$$
____
9 I*\/ 39
x2 = - + --------
4 4
$$x_{2} = \frac{9}{4} + \frac{\sqrt{39} i}{4}$$
x2 = 9/4 + sqrt(39)*i/4
Numerical answer
[src]
x1 = 2.25 - 1.5612494995996*i
x2 = 2.25 + 1.5612494995996*i
x2 = 2.25 + 1.5612494995996*i
The graph