Mister Exam
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- 7x^2+6x+5=0
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7x^2-6x+5=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
This equation is of the form
$$a*x^2 + b*x + c = 0$$
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
$$a = 7$$
$$b = -6$$
$$c = 5$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right) 7 \cdot 4 \cdot 5 + \left(-6\right)^{2} = -104$$
Because D<0, then the equation
has no real roots,
but complex roots is exists.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{3}{7} + \frac{\sqrt{26} i}{7}$$
Simplify
$$x_{2} = \frac{3}{7} - \frac{\sqrt{26} i}{7}$$
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
$$a = 7$$
$$b = -6$$
$$c = 5$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right) 7 \cdot 4 \cdot 5 + \left(-6\right)^{2} = -104$$
Because D<0, then the equation
has no real roots,
but complex roots is exists.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{3}{7} + \frac{\sqrt{26} i}{7}$$
Simplify
$$x_{2} = \frac{3}{7} - \frac{\sqrt{26} i}{7}$$
Simplify
Vieta's Theorem
rewrite the equation
$$7 x^{2} - 6 x + 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{6 x}{7} + \frac{5}{7} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{6}{7}$$
$$q = \frac{c}{a}$$
$$q = \frac{5}{7}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{6}{7}$$
$$x_{1} x_{2} = \frac{5}{7}$$
$$7 x^{2} - 6 x + 5 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{6 x}{7} + \frac{5}{7} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{6}{7}$$
$$q = \frac{c}{a}$$
$$q = \frac{5}{7}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{6}{7}$$
$$x_{1} x_{2} = \frac{5}{7}$$
Rapid solution
[src]
____
3 I*\/ 26
x_1 = - - --------
7 7
$$x_{1} = \frac{3}{7} - \frac{\sqrt{26} i}{7}$$
____
3 I*\/ 26
x_2 = - + --------
7 7
$$x_{2} = \frac{3}{7} + \frac{\sqrt{26} i}{7}$$
Sum and product of roots
[src]
sum
____ ____ 3 I*\/ 26 3 I*\/ 26 - - -------- + - + -------- 7 7 7 7
$$\left(\frac{3}{7} - \frac{\sqrt{26} i}{7}\right) + \left(\frac{3}{7} + \frac{\sqrt{26} i}{7}\right)$$
=
6/7
$$\frac{6}{7}$$
product
____ ____ 3 I*\/ 26 3 I*\/ 26 - - -------- * - + -------- 7 7 7 7
$$\left(\frac{3}{7} - \frac{\sqrt{26} i}{7}\right) * \left(\frac{3}{7} + \frac{\sqrt{26} i}{7}\right)$$
=
5/7
$$\frac{5}{7}$$
Numerical answer
[src]
x1 = 0.428571428571429 - 0.728431359084684*i
x2 = 0.428571428571429 + 0.728431359084684*i
x2 = 0.428571428571429 + 0.728431359084684*i
The graph