Mister exam

# 5x²-8x+3=0 equation

A equation with variable:

#### Numerical solution:

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### The solution

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5*x  - 8*x + 3 = 0
$$\left(5 x^{2} - 8 x\right) + 3 = 0$$
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
$$a = 5$$
$$b = -8$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =

(-8)^2 - 4 * (5) * (3) = 4

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = 1$$
$$x_{2} = \frac{3}{5}$$
The graph
Rapid solution [src]
x1 = 3/5
$$x_{1} = \frac{3}{5}$$
x2 = 1
$$x_{2} = 1$$
x2 = 1
Sum and product of roots [src]
sum
1 + 3/5
$$\frac{3}{5} + 1$$
=
8/5
$$\frac{8}{5}$$
product
3/5
$$\frac{3}{5}$$
=
3/5
$$\frac{3}{5}$$
3/5
Vieta's Theorem
rewrite the equation
$$\left(5 x^{2} - 8 x\right) + 3 = 0$$
of
$$a x^{2} + b x + c = 0$$
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{8 x}{5} + \frac{3}{5} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{8}{5}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{8}{5}$$
$$x_{1} x_{2} = \frac{3}{5}$$
x1 = 1.0
x2 = 0.6
x2 = 0.6