Mister Exam
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15x^2-32x+21=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
$$a = 15$$
$$b = -32$$
$$c = 21$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{16}{15} + \frac{\sqrt{59} i}{15}$$
$$x_{2} = \frac{16}{15} - \frac{\sqrt{59} i}{15}$$
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
$$a = 15$$
$$b = -32$$
$$c = 21$$
, then
D = b^2 - 4 * a * c =
(-32)^2 - 4 * (15) * (21) = -236
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
$$x_{1} = \frac{16}{15} + \frac{\sqrt{59} i}{15}$$
$$x_{2} = \frac{16}{15} - \frac{\sqrt{59} i}{15}$$
Vieta's Theorem
rewrite the equation
$$\left(15 x^{2} - 32 x\right) + 21 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{32 x}{15} + \frac{7}{5} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{32}{15}$$
$$q = \frac{c}{a}$$
$$q = \frac{7}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{32}{15}$$
$$x_{1} x_{2} = \frac{7}{5}$$
$$\left(15 x^{2} - 32 x\right) + 21 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{32 x}{15} + \frac{7}{5} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{32}{15}$$
$$q = \frac{c}{a}$$
$$q = \frac{7}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{32}{15}$$
$$x_{1} x_{2} = \frac{7}{5}$$
Sum and product of roots
[src]
sum
____ ____ 16 I*\/ 59 16 I*\/ 59 -- - -------- + -- + -------- 15 15 15 15
$$\left(\frac{16}{15} - \frac{\sqrt{59} i}{15}\right) + \left(\frac{16}{15} + \frac{\sqrt{59} i}{15}\right)$$
=
32 -- 15
$$\frac{32}{15}$$
product
/ ____\ / ____\ |16 I*\/ 59 | |16 I*\/ 59 | |-- - --------|*|-- + --------| \15 15 / \15 15 /
$$\left(\frac{16}{15} - \frac{\sqrt{59} i}{15}\right) \left(\frac{16}{15} + \frac{\sqrt{59} i}{15}\right)$$
=
7/5
$$\frac{7}{5}$$
7/5
Rapid solution
[src]
____
16 I*\/ 59
x1 = -- - --------
15 15
$$x_{1} = \frac{16}{15} - \frac{\sqrt{59} i}{15}$$
____
16 I*\/ 59
x2 = -- + --------
15 15
$$x_{2} = \frac{16}{15} + \frac{\sqrt{59} i}{15}$$
x2 = 16/15 + sqrt(59)*i/15
Numerical answer
[src]
x1 = 1.06666666666667 + 0.51207638319124*i
x2 = 1.06666666666667 - 0.51207638319124*i
x2 = 1.06666666666667 - 0.51207638319124*i