Mister Exam
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How to use it?
- Equation:
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Equation x+5=0
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Equation 4*x^2-25=0
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Equation 16x^2-1=0
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Identical expressions
- two x^2-9x- one = zero
- 2x squared minus 9x minus 1 equally 0
- two x squared minus 9x minus one equally zero
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- 2x²-9x-1=0
- 2x to the power of 2-9x-1=0
- 2x^2-9x-1=O
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Similar expressions
- 2x^2+9x-1=0
- 2x^2-9x+1=0
2x^2-9x-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 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
$$a = 2$$
$$b = -9$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{9}{4} + \frac{\sqrt{89}}{4}$$
$$x_{2} = \frac{9}{4} - \frac{\sqrt{89}}{4}$$
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
$$a = 2$$
$$b = -9$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(-9)^2 - 4 * (2) * (-1) = 89
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{9}{4} + \frac{\sqrt{89}}{4}$$
$$x_{2} = \frac{9}{4} - \frac{\sqrt{89}}{4}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x^{2} - 9 x\right) - 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{9 x}{2} - \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{9}{2}$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{9}{2}$$
$$x_{1} x_{2} = - \frac{1}{2}$$
$$\left(2 x^{2} - 9 x\right) - 1 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{9 x}{2} - \frac{1}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{9}{2}$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{9}{2}$$
$$x_{1} x_{2} = - \frac{1}{2}$$
Rapid solution
[src]
____
9 \/ 89
x1 = - - ------
4 4
$$x_{1} = \frac{9}{4} - \frac{\sqrt{89}}{4}$$
____
9 \/ 89
x2 = - + ------
4 4
$$x_{2} = \frac{9}{4} + \frac{\sqrt{89}}{4}$$
x2 = 9/4 + sqrt(89)/4
Sum and product of roots
[src]
sum
____ ____ 9 \/ 89 9 \/ 89 - - ------ + - + ------ 4 4 4 4
$$\left(\frac{9}{4} - \frac{\sqrt{89}}{4}\right) + \left(\frac{9}{4} + \frac{\sqrt{89}}{4}\right)$$
=
9/2
$$\frac{9}{2}$$
product
/ ____\ / ____\ |9 \/ 89 | |9 \/ 89 | |- - ------|*|- + ------| \4 4 / \4 4 /
$$\left(\frac{9}{4} - \frac{\sqrt{89}}{4}\right) \left(\frac{9}{4} + \frac{\sqrt{89}}{4}\right)$$
=
-1/2
$$- \frac{1}{2}$$
-1/2