Mister Exam
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How to use it?
- Equation:
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Equation 21-19x+4x^2=x^2-15+2x
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Equation 2x^2+15-3x=11x-5
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Equation 16x+5x^2+12=0
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Equation tg^2(4x)+tg(4x)=0.
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Identical expressions
- twenty-one -19x+4x^ two =x^ two - fifteen +2x
- 21 minus 19x plus 4x squared equally x squared minus 15 plus 2x
- twenty minus one minus 19x plus 4x to the power of two equally x to the power of two minus fifteen plus 2x
- 21-19x+4x2=x2-15+2x
- 21-19x+4x²=x²-15+2x
- 21-19x+4x to the power of 2=x to the power of 2-15+2x
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Similar expressions
- 21-19x+4x^2=x^2-15-2x
- 21+19x+4x^2=x^2-15+2x
- 21-19x+4x^2=x^2+15+2x
- 21-19x-4x^2=x^2-15+2x
21-19x+4x^2=x^2-15+2x equation
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The solution
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[src]
2 2 21 - 19*x + 4*x = x - 15 + 2*x
$$4 x^{2} - 19 x + 21 = x^{2} + 2 x - 15$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$4 x^{2} - 19 x + 21 = x^{2} + 2 x - 15$$
to
$$\left(- x^{2} - 2 x + 15\right) + \left(4 x^{2} - 19 x + 21\right) = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 3$$
$$b = -21$$
$$c = 36$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right) 3 \cdot 4 \cdot 36 + \left(-21\right)^{2} = 9$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = 4$$
Simplify
$$x_{2} = 3$$
Simplify
left part with negative sign.
The equation is transformed from
$$4 x^{2} - 19 x + 21 = x^{2} + 2 x - 15$$
to
$$\left(- x^{2} - 2 x + 15\right) + \left(4 x^{2} - 19 x + 21\right) = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 3$$
$$b = -21$$
$$c = 36$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right) 3 \cdot 4 \cdot 36 + \left(-21\right)^{2} = 9$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = 4$$
Simplify
$$x_{2} = 3$$
Simplify
Vieta's Theorem
rewrite the equation
$$4 x^{2} - 19 x + 21 = x^{2} + 2 x - 15$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 7 x + 12 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = 12$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 7$$
$$x_{1} x_{2} = 12$$
$$4 x^{2} - 19 x + 21 = x^{2} + 2 x - 15$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 7 x + 12 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = 12$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 7$$
$$x_{1} x_{2} = 12$$
Sum and product of roots
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sum
3 + 4
$$\left(3\right) + \left(4\right)$$
=
7
$$7$$
product
3 * 4
$$\left(3\right) * \left(4\right)$$
=
12
$$12$$
The graph