Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation x-y=7
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Equation 1/(x-1)^2+2/(x-1)-3=0
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Equation 1/(x+6)=2
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Equation (x-4)^2-x^2=0
- Express {x} in terms of y where:
- 7*x-1*y=-7
- 10*x-15*y=16
- -14*x-12*y=5
- 13*x-12*y=19
- Graphing y =:
- 2*x^2+6*x+9
- Factor polynomial:
- 2*x^2+6*x+9
- Plot:
- 2*x^2+6*x+9
-
Identical expressions
- two *x^ two + six *x+ nine =3x+ five +(x+ five)=(one -x)- forty
- 2 multiply by x squared plus 6 multiply by x plus 9 equally 3x plus 5 plus (x plus 5) equally (1 minus x) minus 40
- two multiply by x to the power of two plus six multiply by x plus nine equally 3x plus five plus (x plus five) equally (one minus x) minus forty
- 2*x2+6*x+9=3x+5+(x+5)=(1-x)-40
- 2*x2+6*x+9=3x+5+x+5=1-x-40
- 2*x²+6*x+9=3x+5+(x+5)=(1-x)-40
- 2*x to the power of 2+6*x+9=3x+5+(x+5)=(1-x)-40
- 2x^2+6x+9=3x+5+(x+5)=(1-x)-40
- 2x2+6x+9=3x+5+(x+5)=(1-x)-40
- 2x2+6x+9=3x+5+x+5=1-x-40
- 2x^2+6x+9=3x+5+x+5=1-x-40
-
Similar expressions
- 2*x^2+6*x+9=3x+5+(x-5)=(1-x)-40
- 2*x^2+6*x+9=3x+5-(x+5)=(1-x)-40
- 2*x^2+6*x+9=3x+5+(x+5)=(1+x)-40
- 2*x^2+6*x-9=3x+5+(x+5)=(1-x)-40
- 2*x^2-6*x+9=3x+5+(x+5)=(1-x)-40
- 2*x^2+6*x+9=3x-5+(x+5)=(1-x)-40
- 2*x^2+6*x+9=3x+5+(x+5)=(1-x)+40
2*x^2+6*x+9=3x+5+(x+5)=(1-x)-40 equation
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The solution
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[src]
2 2*x + 6*x + 9 = 3*x + 5 + x + 5
$$\left(2 x^{2} + 6 x\right) + 9 = \left(x + 5\right) + \left(3 x + 5\right)$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(2 x^{2} + 6 x\right) + 9 = \left(x + 5\right) + \left(3 x + 5\right)$$
to
$$\left(\left(- 3 x - 5\right) + \left(- x - 5\right)\right) + \left(\left(2 x^{2} + 6 x\right) + 9\right) = 0$$
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
$$a = 2$$
$$b = 2$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{3}}{2}$$
$$x_{2} = - \frac{\sqrt{3}}{2} - \frac{1}{2}$$
left part with negative sign.
The equation is transformed from
$$\left(2 x^{2} + 6 x\right) + 9 = \left(x + 5\right) + \left(3 x + 5\right)$$
to
$$\left(\left(- 3 x - 5\right) + \left(- x - 5\right)\right) + \left(\left(2 x^{2} + 6 x\right) + 9\right) = 0$$
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
$$a = 2$$
$$b = 2$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(2)^2 - 4 * (2) * (-1) = 12
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{3}}{2}$$
$$x_{2} = - \frac{\sqrt{3}}{2} - \frac{1}{2}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x^{2} + 6 x\right) + 9 = \left(x + 5\right) + \left(3 x + 5\right)$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + x - \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = - \frac{1}{2}$$
$$\left(2 x^{2} + 6 x\right) + 9 = \left(x + 5\right) + \left(3 x + 5\right)$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + x - \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = - \frac{1}{2}$$
Sum and product of roots
[src]
sum
___ ___ 1 \/ 3 1 \/ 3 - - + ----- + - - - ----- 2 2 2 2
$$\left(- \frac{\sqrt{3}}{2} - \frac{1}{2}\right) + \left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right)$$
=
-1
$$-1$$
product
/ ___\ / ___\ | 1 \/ 3 | | 1 \/ 3 | |- - + -----|*|- - - -----| \ 2 2 / \ 2 2 /
$$\left(- \frac{1}{2} + \frac{\sqrt{3}}{2}\right) \left(- \frac{\sqrt{3}}{2} - \frac{1}{2}\right)$$
=
-1/2
$$- \frac{1}{2}$$
-1/2
Rapid solution
[src]
___
1 \/ 3
x1 = - - + -----
2 2
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{3}}{2}$$
___
1 \/ 3
x2 = - - - -----
2 2
$$x_{2} = - \frac{\sqrt{3}}{2} - \frac{1}{2}$$
x2 = -sqrt(3)/2 - 1/2