Mister Exam
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Identical expressions
- x^ two - sixty-seven *x+ four hundred and five = zero
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- x2-67*x+405=0
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- x^2-67x+405=0
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Similar expressions
- x^2+67*x+405=0
- x^2-67*x-405=0
x^2-67*x+405=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
$$a = 1$$
$$b = -67$$
$$c = 405$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{\sqrt{2869}}{2} + \frac{67}{2}$$
$$x_{2} = \frac{67}{2} - \frac{\sqrt{2869}}{2}$$
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
$$a = 1$$
$$b = -67$$
$$c = 405$$
, then
D = b^2 - 4 * a * c =
(-67)^2 - 4 * (1) * (405) = 2869
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{2869}}{2} + \frac{67}{2}$$
$$x_{2} = \frac{67}{2} - \frac{\sqrt{2869}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -67$$
$$q = \frac{c}{a}$$
$$q = 405$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 67$$
$$x_{1} x_{2} = 405$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -67$$
$$q = \frac{c}{a}$$
$$q = 405$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 67$$
$$x_{1} x_{2} = 405$$
Rapid solution
[src]
______
67 \/ 2869
x1 = -- - --------
2 2
$$x_{1} = \frac{67}{2} - \frac{\sqrt{2869}}{2}$$
______
67 \/ 2869
x2 = -- + --------
2 2
$$x_{2} = \frac{\sqrt{2869}}{2} + \frac{67}{2}$$
x2 = sqrt(2869)/2 + 67/2
Sum and product of roots
[src]
sum
______ ______ 67 \/ 2869 67 \/ 2869 -- - -------- + -- + -------- 2 2 2 2
$$\left(\frac{67}{2} - \frac{\sqrt{2869}}{2}\right) + \left(\frac{\sqrt{2869}}{2} + \frac{67}{2}\right)$$
=
67
$$67$$
product
/ ______\ / ______\ |67 \/ 2869 | |67 \/ 2869 | |-- - --------|*|-- + --------| \2 2 / \2 2 /
$$\left(\frac{67}{2} - \frac{\sqrt{2869}}{2}\right) \left(\frac{\sqrt{2869}}{2} + \frac{67}{2}\right)$$
=
405
$$405$$
405