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14-4x^2-x=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
$$a = -4$$
$$b = -1$$
$$c = 14$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right)^{2} - \left(-4\right) 4 \cdot 14 = 225$$
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
$$x_{1} = -2$$
Simplify
$$x_{2} = \frac{7}{4}$$
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
$$a = -4$$
$$b = -1$$
$$c = 14$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right)^{2} - \left(-4\right) 4 \cdot 14 = 225$$
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
$$x_{1} = -2$$
Simplify
$$x_{2} = \frac{7}{4}$$
Simplify
Vieta's Theorem
rewrite the equation
$$- 4 x^{2} - x + 14 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{x}{4} - \frac{7}{2} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{1}{4}$$
$$q = \frac{c}{a}$$
$$q = - \frac{7}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{1}{4}$$
$$x_{1} x_{2} = - \frac{7}{2}$$
$$- 4 x^{2} - x + 14 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{x}{4} - \frac{7}{2} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{1}{4}$$
$$q = \frac{c}{a}$$
$$q = - \frac{7}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{1}{4}$$
$$x_{1} x_{2} = - \frac{7}{2}$$
Sum and product of roots
[src]
sum
-2 + 7/4
$$\left(-2\right) + \left(\frac{7}{4}\right)$$
=
-1/4
$$- \frac{1}{4}$$
product
-2 * 7/4
$$\left(-2\right) * \left(\frac{7}{4}\right)$$
=
-7/2
$$- \frac{7}{2}$$
The graph