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6x-8x²+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
$$a = -8$$
$$b = 6$$
$$c = 5$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{1}{2}$$
$$x_{2} = \frac{5}{4}$$
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
$$a = -8$$
$$b = 6$$
$$c = 5$$
, then
D = b^2 - 4 * a * c =
(6)^2 - 4 * (-8) * (5) = 196
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{1}{2}$$
$$x_{2} = \frac{5}{4}$$
Vieta's Theorem
rewrite the equation
$$\left(- 8 x^{2} + 6 x\right) + 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{3 x}{4} - \frac{5}{8} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{3}{4}$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{8}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{3}{4}$$
$$x_{1} x_{2} = - \frac{5}{8}$$
$$\left(- 8 x^{2} + 6 x\right) + 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{3 x}{4} - \frac{5}{8} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{3}{4}$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{8}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{3}{4}$$
$$x_{1} x_{2} = - \frac{5}{8}$$
Sum and product of roots
[src]
sum
-1/2 + 5/4
$$- \frac{1}{2} + \frac{5}{4}$$
=
3/4
$$\frac{3}{4}$$
product
-5 --- 2*4
$$- \frac{5}{8}$$
=
-5/8
$$- \frac{5}{8}$$
-5/8