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Equation 2*x^2+2*x-12=0
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Equation (x-17)^3=16*(x-17)
- Express {x} in terms of y where:
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- Graphing y =:
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- Derivative of:
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2x²
- Integral of d{x}:
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2x²
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Identical expressions
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Similar expressions
- (x-4²)-(x+9²)=2x²
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- (x²+2x)²-(X+1)²=55
- (x+4²)+(x+9²)=2x²
(x-4²)+(x+9²)=2x² equation
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The solution
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[src]
2 x - 16 + x + 81 = 2*x
$$\left(x - 16\right) + \left(x + 81\right) = 2 x^{2}$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(x - 16\right) + \left(x + 81\right) = 2 x^{2}$$
to
$$- 2 x^{2} + \left(\left(x - 16\right) + \left(x + 81\right)\right) = 0$$
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
$$a = -2$$
$$b = 2$$
$$c = 65$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{1}{2} - \frac{\sqrt{131}}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{131}}{2}$$
left part with negative sign.
The equation is transformed from
$$\left(x - 16\right) + \left(x + 81\right) = 2 x^{2}$$
to
$$- 2 x^{2} + \left(\left(x - 16\right) + \left(x + 81\right)\right) = 0$$
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 - it is the discriminant.
Because
$$a = -2$$
$$b = 2$$
$$c = 65$$
, then
D = b^2 - 4 * a * c =
(2)^2 - 4 * (-2) * (65) = 524
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{2} - \frac{\sqrt{131}}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{131}}{2}$$
Vieta's Theorem
rewrite the equation
$$\left(x - 16\right) + \left(x + 81\right) = 2 x^{2}$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - x - \frac{65}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = - \frac{65}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = - \frac{65}{2}$$
$$\left(x - 16\right) + \left(x + 81\right) = 2 x^{2}$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - x - \frac{65}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = - \frac{65}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = - \frac{65}{2}$$
Sum and product of roots
[src]
sum
_____ _____ 1 \/ 131 1 \/ 131 - - ------- + - + ------- 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{131}}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{131}}{2}\right)$$
=
1
$$1$$
product
/ _____\ / _____\ |1 \/ 131 | |1 \/ 131 | |- - -------|*|- + -------| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{131}}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{131}}{2}\right)$$
=
-65/2
$$- \frac{65}{2}$$
-65/2
Rapid solution
[src]
_____
1 \/ 131
x1 = - - -------
2 2
$$x_{1} = \frac{1}{2} - \frac{\sqrt{131}}{2}$$
_____
1 \/ 131
x2 = - + -------
2 2
$$x_{2} = \frac{1}{2} + \frac{\sqrt{131}}{2}$$
x2 = 1/2 + sqrt(131)/2