Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x^2+3x+2=0
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Equation z^2-6*z+10=0
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Equation x^2+5x+6=0
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Equation -x/7+x=15/7
- Express {x} in terms of y where:
- -20*x-6*y=4
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Identical expressions
- a^ two *x^ two +a*x- twenty-one *a+ one = zero
- a squared multiply by x squared plus a multiply by x minus 21 multiply by a plus 1 equally 0
- a to the power of two multiply by x to the power of two plus a multiply by x minus twenty minus one multiply by a plus one equally zero
- a2*x2+a*x-21*a+1=0
- a²*x²+a*x-21*a+1=0
- a to the power of 2*x to the power of 2+a*x-21*a+1=0
- a^2x^2+ax-21a+1=0
- a2x2+ax-21a+1=0
- a^2*x^2+a*x-21*a+1=O
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Similar expressions
- a^2*x^2+a*x-21*a-1=0
- a^2*x^2-a*x-21*a+1=0
- a^2*x^2+a*x+21*a+1=0
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What you mean?
- a^2*x^2 + a*x - 21*a + 1 = 0
- a*x - 21*a + a^((2*x)^2) + 1 = 0
- a*x - 21*a + a^((2*x)^2) + 1 = 0
You entered:
a^2*x^2+a*x-21*a+1=0
What you mean?
a^2*x^2+a*x-21*a+1=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
$$a = a^{2}$$
$$b = a$$
$$c = - 21 a + 1$$
, then
$$D = b^2 - 4\ a\ c = $$
$$a^{2} - 4 a^{2} \cdot \left(- 21 a + 1\right) = - 4 a^{2} \cdot \left(- 21 a + 1\right) + a^{2}$$
The equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{- a + \sqrt{- 4 a^{2} \cdot \left(- 21 a + 1\right) + a^{2}}}{2 a^{2}}$$
Simplify
$$x_{2} = \frac{- a - \sqrt{- 4 a^{2} \cdot \left(- 21 a + 1\right) + a^{2}}}{2 a^{2}}$$
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
$$a = a^{2}$$
$$b = a$$
$$c = - 21 a + 1$$
, then
$$D = b^2 - 4\ a\ c = $$
$$a^{2} - 4 a^{2} \cdot \left(- 21 a + 1\right) = - 4 a^{2} \cdot \left(- 21 a + 1\right) + a^{2}$$
The equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{- a + \sqrt{- 4 a^{2} \cdot \left(- 21 a + 1\right) + a^{2}}}{2 a^{2}}$$
Simplify
$$x_{2} = \frac{- a - \sqrt{- 4 a^{2} \cdot \left(- 21 a + 1\right) + a^{2}}}{2 a^{2}}$$
Simplify
The solution of the parametric equation
Given the equation with a parameter:
$$a^{2} x^{2} + a x - 21 a + 1 = 0$$
The coefficient at x is equal to
$$a^{2}$$
then possible cases for a :
$$a < 0$$
$$a = 0$$
Consider all cases in more detail:
With
$$a < 0$$
the equation
$$x^{2} - x + 22 = 0$$
its solution
With
$$a = 0$$
the equation
$$1 = 0$$
its solution
no solutions
$$a^{2} x^{2} + a x - 21 a + 1 = 0$$
The coefficient at x is equal to
$$a^{2}$$
then possible cases for a :
$$a < 0$$
$$a = 0$$
Consider all cases in more detail:
With
$$a < 0$$
the equation
$$x^{2} - x + 22 = 0$$
its solution
With
$$a = 0$$
the equation
$$1 = 0$$
its solution
no solutions
Vieta's Theorem
rewrite the equation
$$a^{2} x^{2} + a x - 21 a + 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a^{2} x^{2} + a x - 21 a + 1}{a^{2}} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{1}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{\left(-1\right) 21 a + 1}{a^{2}}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{1}{a}$$
$$x_{1} x_{2} = \frac{\left(-1\right) 21 a + 1}{a^{2}}$$
$$a^{2} x^{2} + a x - 21 a + 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a^{2} x^{2} + a x - 21 a + 1}{a^{2}} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{1}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{\left(-1\right) 21 a + 1}{a^{2}}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{1}{a}$$
$$x_{1} x_{2} = \frac{\left(-1\right) 21 a + 1}{a^{2}}$$
The graph
Sum and product of roots
[src]
sum
___________ ___________
1 \/ -3 + 84*a 1 \/ -3 + 84*a
- - - ------------- - - + -------------
2 2 2 2
------------------- + -------------------
a a
$$\left(\frac{- \frac{\sqrt{84 a - 3}}{2} - \frac{1}{2}}{a}\right) + \left(\frac{\frac{\sqrt{84 a - 3}}{2} - \frac{1}{2}}{a}\right)$$
=
___________ ___________
1 \/ -3 + 84*a 1 \/ -3 + 84*a
- - + ------------- - - - -------------
2 2 2 2
------------------- + -------------------
a a
$$\frac{- \frac{\sqrt{84 a - 3}}{2} - \frac{1}{2}}{a} + \frac{\frac{\sqrt{84 a - 3}}{2} - \frac{1}{2}}{a}$$
product
___________ ___________
1 \/ -3 + 84*a 1 \/ -3 + 84*a
- - - ------------- - - + -------------
2 2 2 2
------------------- * -------------------
a a
$$\left(\frac{- \frac{\sqrt{84 a - 3}}{2} - \frac{1}{2}}{a}\right) * \left(\frac{\frac{\sqrt{84 a - 3}}{2} - \frac{1}{2}}{a}\right)$$
=
1 - 21*a
--------
2
a
$$\frac{- 21 a + 1}{a^{2}}$$
Rapid solution
[src]
___________
1 \/ -3 + 84*a
- - - -------------
2 2
x_1 = -------------------
a
$$x_{1} = \frac{- \frac{\sqrt{84 a - 3}}{2} - \frac{1}{2}}{a}$$
___________
1 \/ -3 + 84*a
- - + -------------
2 2
x_2 = -------------------
a
$$x_{2} = \frac{\frac{\sqrt{84 a - 3}}{2} - \frac{1}{2}}{a}$$