Mister Exam
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-0.25x^2+x-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Expand the expression in the equation
$$\left(- \frac{x^{2}}{4} + x - 1\right) + 0 = 0$$
We get the quadratic equation
$$- \frac{x^{2}}{4} + x - 1 = 0$$
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
$$a = - \frac{1}{4}$$
$$b = 1$$
$$c = -1$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right) \left(\left(- \frac{1}{4}\right) 4\right) \left(-1\right) + 1^{2} = 0$$
Because D = 0, then the equation has one root.
$$x_{1} = 2$$
$$\left(- \frac{x^{2}}{4} + x - 1\right) + 0 = 0$$
We get the quadratic equation
$$- \frac{x^{2}}{4} + x - 1 = 0$$
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
$$a = - \frac{1}{4}$$
$$b = 1$$
$$c = -1$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right) \left(\left(- \frac{1}{4}\right) 4\right) \left(-1\right) + 1^{2} = 0$$
Because D = 0, then the equation has one root.
x = -b/2a = -1/2/(-1/4)
$$x_{1} = 2$$
Vieta's Theorem
rewrite the equation
$$- \frac{x^{2}}{4} + x - 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x + 4 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = 4$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = 4$$
$$- \frac{x^{2}}{4} + x - 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x + 4 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = 4$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = 4$$
Sum and product of roots
[src]
sum
2
$$\left(2\right)$$
=
2
$$2$$
product
2
$$\left(2\right)$$
=
2
$$2$$
The graph