Mister Exam
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Similar expressions
- x^2-5x+12=0
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x^2-5x-12=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = -5$$
$$c = -12$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-5\right)^{2} - 1 \cdot 4 \left(-12\right) = 73$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{73}}{2}$$
Simplify
$$x_{2} = - \frac{\sqrt{73}}{2} + \frac{5}{2}$$
Simplify
$$a\ x^2 + b\ x + c = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = -5$$
$$c = -12$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-5\right)^{2} - 1 \cdot 4 \left(-12\right) = 73$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{73}}{2}$$
Simplify
$$x_{2} = - \frac{\sqrt{73}}{2} + \frac{5}{2}$$
Simplify
Vieta's Theorem
it is reduced quadratic equation
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -12$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = -12$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -12$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = -12$$
Rapid solution
[src]
____
5 \/ 73
x_1 = - - ------
2 2
$$x_{1} = - \frac{\sqrt{73}}{2} + \frac{5}{2}$$
____
5 \/ 73
x_2 = - + ------
2 2
$$x_{2} = \frac{5}{2} + \frac{\sqrt{73}}{2}$$
Sum and product of roots
[src]
sum
____ ____ 5 \/ 73 5 \/ 73 - - ------ + - + ------ 2 2 2 2
$$\left(- \frac{\sqrt{73}}{2} + \frac{5}{2}\right) + \left(\frac{5}{2} + \frac{\sqrt{73}}{2}\right)$$
=
5
$$5$$
product
____ ____ 5 \/ 73 5 \/ 73 - - ------ * - + ------ 2 2 2 2
$$\left(- \frac{\sqrt{73}}{2} + \frac{5}{2}\right) * \left(\frac{5}{2} + \frac{\sqrt{73}}{2}\right)$$
=
-12
$$-12$$
The graph