Mister Exam

Other calculators

15x^2-32x+21=0 equation

The teacher will be very surprised to see your correct solution 😉

v

Numerical solution:

Do search numerical solution at [, ]

The solution

You have entered [src]
    2                
15*x  - 32*x + 21 = 0
$$\left(15 x^{2} - 32 x\right) + 21 = 0$$
Detail solution
This equation is of the form
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
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{32}{15}$$
$$x_{1} x_{2} = \frac{7}{5}$$
The graph
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