Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 5-x=4(x-3)
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Equation 2*x^2+5*x-25=13*x+17
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- -16*x-20*y=-13
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Identical expressions
- forty-five (x+ one)^ two =243x
- 45(x plus 1) squared equally 243x
- forty minus five (x plus one) to the power of two equally 243x
- 45(x+1)2=243x
- 45x+12=243x
- 45(x+1)²=243x
- 45(x+1) to the power of 2=243x
- 45x+1^2=243x
-
Similar expressions
- 3*243*x^3/10+2*x^2/(13/50)+x=27*3/1000+943/1000
- 243*x^4-6*x^3-1656*x^2+320/3*x+1280=0
- 45(x-1)^2=243x
45(x+1)^2=243x equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$45 \left(x + 1\right)^{2} = 243 x$$
to
$$- 243 x + 45 \left(x + 1\right)^{2} = 0$$
Expand the expression in the equation
$$- 243 x + 45 \left(x + 1\right)^{2} = 0$$
We get the quadratic equation
$$45 x^{2} - 153 x + 45 = 0$$
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
$$a = 45$$
$$b = -153$$
$$c = 45$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{3 \sqrt{21}}{10} + \frac{17}{10}$$
$$x_{2} = \frac{17}{10} - \frac{3 \sqrt{21}}{10}$$
left part with negative sign.
The equation is transformed from
$$45 \left(x + 1\right)^{2} = 243 x$$
to
$$- 243 x + 45 \left(x + 1\right)^{2} = 0$$
Expand the expression in the equation
$$- 243 x + 45 \left(x + 1\right)^{2} = 0$$
We get the quadratic equation
$$45 x^{2} - 153 x + 45 = 0$$
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
$$a = 45$$
$$b = -153$$
$$c = 45$$
, then
D = b^2 - 4 * a * c =
(-153)^2 - 4 * (45) * (45) = 15309
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{3 \sqrt{21}}{10} + \frac{17}{10}$$
$$x_{2} = \frac{17}{10} - \frac{3 \sqrt{21}}{10}$$
Rapid solution
[src]
____
17 3*\/ 21
x1 = -- - --------
10 10
$$x_{1} = \frac{17}{10} - \frac{3 \sqrt{21}}{10}$$
____
17 3*\/ 21
x2 = -- + --------
10 10
$$x_{2} = \frac{3 \sqrt{21}}{10} + \frac{17}{10}$$
x2 = 3*sqrt(21)/10 + 17/10
Sum and product of roots
[src]
sum
____ ____ 17 3*\/ 21 17 3*\/ 21 -- - -------- + -- + -------- 10 10 10 10
$$\left(\frac{17}{10} - \frac{3 \sqrt{21}}{10}\right) + \left(\frac{3 \sqrt{21}}{10} + \frac{17}{10}\right)$$
=
17/5
$$\frac{17}{5}$$
product
/ ____\ / ____\ |17 3*\/ 21 | |17 3*\/ 21 | |-- - --------|*|-- + --------| \10 10 / \10 10 /
$$\left(\frac{17}{10} - \frac{3 \sqrt{21}}{10}\right) \left(\frac{3 \sqrt{21}}{10} + \frac{17}{10}\right)$$
=
1
$$1$$
1