Mister Exam
(d+7)(-d-7)+(2d-1) equation
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The solution
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(d + 7)*(-d - 7) + 2*d - 1 = 0
$$\left(- d - 7\right) \left(d + 7\right) + \left(2 d - 1\right) = 0$$
Detail solution
Expand the expression in the equation
$$\left(- d - 7\right) \left(d + 7\right) + \left(2 d - 1\right) = 0$$
We get the quadratic equation
$$- d^{2} - 12 d - 50 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$d_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$d_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -1$$
$$b = -12$$
$$c = -50$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$d_{1} = -6 - \sqrt{14} i$$
$$d_{2} = -6 + \sqrt{14} i$$
$$\left(- d - 7\right) \left(d + 7\right) + \left(2 d - 1\right) = 0$$
We get the quadratic equation
$$- d^{2} - 12 d - 50 = 0$$
This equation is of the form
a*d^2 + b*d + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$d_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$d_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -1$$
$$b = -12$$
$$c = -50$$
, then
D = b^2 - 4 * a * c =
(-12)^2 - 4 * (-1) * (-50) = -56
Because D<0, then the equation
has no real roots,
but complex roots is exists.
d1 = (-b + sqrt(D)) / (2*a)
d2 = (-b - sqrt(D)) / (2*a)
or
$$d_{1} = -6 - \sqrt{14} i$$
$$d_{2} = -6 + \sqrt{14} i$$
Rapid solution
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____ d1 = -6 - I*\/ 14
$$d_{1} = -6 - \sqrt{14} i$$
____ d2 = -6 + I*\/ 14
$$d_{2} = -6 + \sqrt{14} i$$
d2 = -6 + sqrt(14)*i
Sum and product of roots
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sum
____ ____ -6 - I*\/ 14 + -6 + I*\/ 14
$$\left(-6 - \sqrt{14} i\right) + \left(-6 + \sqrt{14} i\right)$$
=
-12
$$-12$$
product
/ ____\ / ____\ \-6 - I*\/ 14 /*\-6 + I*\/ 14 /
$$\left(-6 - \sqrt{14} i\right) \left(-6 + \sqrt{14} i\right)$$
=
50
$$50$$
50
Numerical answer
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d1 = -6.0 - 3.74165738677394*i
d2 = -6.0 + 3.74165738677394*i
d2 = -6.0 + 3.74165738677394*i