Mister Exam
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Identical expressions
- 4x^ two -1 zero x- five =0
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Similar expressions
- 4x^2+10x-5=0
- 4x^2-10x+5=0
4x^2-10x-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
$$a = 4$$
$$b = -10$$
$$c = -5$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{5}{4} + \frac{3 \sqrt{5}}{4}$$
$$x_{2} = \frac{5}{4} - \frac{3 \sqrt{5}}{4}$$
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
$$a = 4$$
$$b = -10$$
$$c = -5$$
, then
D = b^2 - 4 * a * c =
(-10)^2 - 4 * (4) * (-5) = 180
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{5}{4} + \frac{3 \sqrt{5}}{4}$$
$$x_{2} = \frac{5}{4} - \frac{3 \sqrt{5}}{4}$$
Vieta's Theorem
rewrite the equation
$$\left(4 x^{2} - 10 x\right) - 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{5 x}{2} - \frac{5}{4} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{5}{2}$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{4}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{5}{2}$$
$$x_{1} x_{2} = - \frac{5}{4}$$
$$\left(4 x^{2} - 10 x\right) - 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{5 x}{2} - \frac{5}{4} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{5}{2}$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{4}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{5}{2}$$
$$x_{1} x_{2} = - \frac{5}{4}$$
Sum and product of roots
[src]
sum
___ ___ 5 3*\/ 5 5 3*\/ 5 - - ------- + - + ------- 4 4 4 4
$$\left(\frac{5}{4} - \frac{3 \sqrt{5}}{4}\right) + \left(\frac{5}{4} + \frac{3 \sqrt{5}}{4}\right)$$
=
5/2
$$\frac{5}{2}$$
product
/ ___\ / ___\ |5 3*\/ 5 | |5 3*\/ 5 | |- - -------|*|- + -------| \4 4 / \4 4 /
$$\left(\frac{5}{4} - \frac{3 \sqrt{5}}{4}\right) \left(\frac{5}{4} + \frac{3 \sqrt{5}}{4}\right)$$
=
-5/4
$$- \frac{5}{4}$$
-5/4
Rapid solution
[src]
___
5 3*\/ 5
x1 = - - -------
4 4
$$x_{1} = \frac{5}{4} - \frac{3 \sqrt{5}}{4}$$
___
5 3*\/ 5
x2 = - + -------
4 4
$$x_{2} = \frac{5}{4} + \frac{3 \sqrt{5}}{4}$$
x2 = 5/4 + 3*sqrt(5)/4