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How to use it?
- Equation:
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Equation (x-3)*(x+4)=0
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Equation x^3-x^2-5=0
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Equation x-2(3x+2)=16
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Identical expressions
- ax^ two + eight *x- four *a- sixteen = zero
- ax squared plus 8 multiply by x minus 4 multiply by a minus 16 equally 0
- ax to the power of two plus eight multiply by x minus four multiply by a minus sixteen equally zero
- ax2+8*x-4*a-16=0
- ax²+8*x-4*a-16=0
- ax to the power of 2+8*x-4*a-16=0
- ax^2+8x-4a-16=0
- ax2+8x-4a-16=0
- ax^2+8*x-4*a-16=O
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Similar expressions
- ax^2+8*x-4*a+16=0
- ax^2-8*x-4*a-16=0
- ax^2+8*x+4*a-16=0
ax^2+8*x-4*a-16=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
2 a*x + 8*x - 4*a - 16 = 0
$$\left(- 4 a + \left(a x^{2} + 8 x\right)\right) - 16 = 0$$
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
$$b = 8$$
$$c = - 4 a - 16$$
, then
The equation has two roots.
or
$$x_{1} = \frac{\sqrt{- 4 a \left(- 4 a - 16\right) + 64} - 8}{2 a}$$
$$x_{2} = \frac{- \sqrt{- 4 a \left(- 4 a - 16\right) + 64} - 8}{2 a}$$
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
True
$$b = 8$$
$$c = - 4 a - 16$$
, then
D = b^2 - 4 * a * c =
(8)^2 - 4 * (a) * (-16 - 4*a) = 64 - 4*a*(-16 - 4*a)
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{- 4 a \left(- 4 a - 16\right) + 64} - 8}{2 a}$$
$$x_{2} = \frac{- \sqrt{- 4 a \left(- 4 a - 16\right) + 64} - 8}{2 a}$$
The solution of the parametric equation
Given the equation with a parameter:
$$a x^{2} - 4 a + 8 x - 16 = 0$$
Коэффициент при x равен
$$a$$
then possible cases for a :
$$a < 0$$
$$a = 0$$
Consider all cases in more detail:
With
$$a < 0$$
the equation
$$- x^{2} + 8 x - 12 = 0$$
its solution
$$x = 2$$
$$x = 6$$
With
$$a = 0$$
the equation
$$8 x - 16 = 0$$
its solution
$$x = 2$$
$$a x^{2} - 4 a + 8 x - 16 = 0$$
Коэффициент при x равен
$$a$$
then possible cases for a :
$$a < 0$$
$$a = 0$$
Consider all cases in more detail:
With
$$a < 0$$
the equation
$$- x^{2} + 8 x - 12 = 0$$
its solution
$$x = 2$$
$$x = 6$$
With
$$a = 0$$
the equation
$$8 x - 16 = 0$$
its solution
$$x = 2$$
Vieta's Theorem
rewrite the equation
$$\left(- 4 a + \left(a x^{2} + 8 x\right)\right) - 16 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} - 4 a + 8 x - 16}{a} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{8}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{- 4 a - 16}{a}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{8}{a}$$
$$x_{1} x_{2} = \frac{- 4 a - 16}{a}$$
$$\left(- 4 a + \left(a x^{2} + 8 x\right)\right) - 16 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} - 4 a + 8 x - 16}{a} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{8}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{- 4 a - 16}{a}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{8}{a}$$
$$x_{1} x_{2} = \frac{- 4 a - 16}{a}$$
The graph
Rapid solution
[src]
x1 = 2
$$x_{1} = 2$$
8*re(a) 8*I*im(a)
x2 = -2 - --------------- + ---------------
2 2 2 2
im (a) + re (a) im (a) + re (a)
$$x_{2} = -2 - \frac{8 \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{8 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
x2 = -2 - 8*re(a)/(re(a)^2 + im(a)^2) + 8*i*im(a)/(re(a)^2 + im(a)^2)
Sum and product of roots
[src]
sum
8*re(a) 8*I*im(a)
2 + -2 - --------------- + ---------------
2 2 2 2
im (a) + re (a) im (a) + re (a)
$$\left(-2 - \frac{8 \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{8 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + 2$$
=
8*re(a) 8*I*im(a)
- --------------- + ---------------
2 2 2 2
im (a) + re (a) im (a) + re (a)
$$- \frac{8 \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{8 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
product
/ 8*re(a) 8*I*im(a) \ 2*|-2 - --------------- + ---------------| | 2 2 2 2 | \ im (a) + re (a) im (a) + re (a)/
$$2 \left(-2 - \frac{8 \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{8 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right)$$
=
/ 2 2 \
4*\- im (a) - re (a) - 4*re(a) + 4*I*im(a)/
-------------------------------------------
2 2
im (a) + re (a)
$$\frac{4 \left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} - 4 \operatorname{re}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 4 i \operatorname{im}{\left(a\right)}\right)}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
4*(-im(a)^2 - re(a)^2 - 4*re(a) + 4*i*im(a))/(im(a)^2 + re(a)^2)