Mister Exam
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Equation 9+12x-5x^2=0
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9+12x-5x^2=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
$$a = -5$$
$$b = 12$$
$$c = 9$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{3}{5}$$
$$x_{2} = 3$$
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
$$a = -5$$
$$b = 12$$
$$c = 9$$
, then
D = b^2 - 4 * a * c =
(12)^2 - 4 * (-5) * (9) = 324
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{3}{5}$$
$$x_{2} = 3$$
Vieta's Theorem
rewrite the equation
$$- 5 x^{2} + \left(12 x + 9\right) = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{12 x}{5} - \frac{9}{5} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{12}{5}$$
$$q = \frac{c}{a}$$
$$q = - \frac{9}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{12}{5}$$
$$x_{1} x_{2} = - \frac{9}{5}$$
$$- 5 x^{2} + \left(12 x + 9\right) = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{12 x}{5} - \frac{9}{5} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{12}{5}$$
$$q = \frac{c}{a}$$
$$q = - \frac{9}{5}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{12}{5}$$
$$x_{1} x_{2} = - \frac{9}{5}$$
Sum and product of roots
[src]
sum
3 - 3/5
$$- \frac{3}{5} + 3$$
=
12/5
$$\frac{12}{5}$$
product
3*(-3) ------ 5
$$\frac{\left(-3\right) 3}{5}$$
=
-9/5
$$- \frac{9}{5}$$
-9/5
The graph