Mister Exam
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How to use it?
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Identical expressions
- three *x^ two - ten *x+ five = zero
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Similar expressions
- 3*x^2+10*x+5=0
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3*x^2-10*x+5=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
$$a = 3$$
$$b = -10$$
$$c = 5$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{\sqrt{10}}{3} + \frac{5}{3}$$
$$x_{2} = \frac{5}{3} - \frac{\sqrt{10}}{3}$$
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
$$a = 3$$
$$b = -10$$
$$c = 5$$
, then
D = b^2 - 4 * a * c =
(-10)^2 - 4 * (3) * (5) = 40
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{10}}{3} + \frac{5}{3}$$
$$x_{2} = \frac{5}{3} - \frac{\sqrt{10}}{3}$$
Vieta's Theorem
rewrite the equation
$$\left(3 x^{2} - 10 x\right) + 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{10 x}{3} + \frac{5}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{10}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{5}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{10}{3}$$
$$x_{1} x_{2} = \frac{5}{3}$$
$$\left(3 x^{2} - 10 x\right) + 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{10 x}{3} + \frac{5}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{10}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{5}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{10}{3}$$
$$x_{1} x_{2} = \frac{5}{3}$$
Sum and product of roots
[src]
sum
____ ____ 5 \/ 10 5 \/ 10 - - ------ + - + ------ 3 3 3 3
$$\left(\frac{5}{3} - \frac{\sqrt{10}}{3}\right) + \left(\frac{\sqrt{10}}{3} + \frac{5}{3}\right)$$
=
10/3
$$\frac{10}{3}$$
product
/ ____\ / ____\ |5 \/ 10 | |5 \/ 10 | |- - ------|*|- + ------| \3 3 / \3 3 /
$$\left(\frac{5}{3} - \frac{\sqrt{10}}{3}\right) \left(\frac{\sqrt{10}}{3} + \frac{5}{3}\right)$$
=
5/3
$$\frac{5}{3}$$
5/3
Rapid solution
[src]
____
5 \/ 10
x1 = - - ------
3 3
$$x_{1} = \frac{5}{3} - \frac{\sqrt{10}}{3}$$
____
5 \/ 10
x2 = - + ------
3 3
$$x_{2} = \frac{\sqrt{10}}{3} + \frac{5}{3}$$
x2 = sqrt(10)/3 + 5/3