Mister Exam
Factor (x^2-5*x-84)/24 squared
The solution
You have entered
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2
x - 5*x - 84
-------------
24
$$\frac{\left(x^{2} - 5 x\right) - 84}{24}$$
(x^2 - 5*x - 84)/24
Fraction decomposition
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-7/2 - 5*x/24 + x^2/24
$$\frac{x^{2}}{24} - \frac{5 x}{24} - \frac{7}{2}$$
2 7 5*x x - - - --- + -- 2 24 24
General simplification
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2 7 5*x x - - - --- + -- 2 24 24
$$\frac{x^{2}}{24} - \frac{5 x}{24} - \frac{7}{2}$$
-7/2 - 5*x/24 + x^2/24
The perfect square
Let's highlight the perfect square of the square three-member
$$\frac{\left(x^{2} - 5 x\right) - 84}{24}$$
To do this, let's use the formula
$$a x^{2} + b x + c = a \left(m + x\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = \frac{1}{24}$$
$$b = - \frac{5}{24}$$
$$c = - \frac{7}{2}$$
Then
$$m = - \frac{5}{2}$$
$$n = - \frac{361}{96}$$
So,
$$\frac{\left(x - \frac{5}{2}\right)^{2}}{24} - \frac{361}{96}$$
$$\frac{\left(x^{2} - 5 x\right) - 84}{24}$$
To do this, let's use the formula
$$a x^{2} + b x + c = a \left(m + x\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = \frac{1}{24}$$
$$b = - \frac{5}{24}$$
$$c = - \frac{7}{2}$$
Then
$$m = - \frac{5}{2}$$
$$n = - \frac{361}{96}$$
So,
$$\frac{\left(x - \frac{5}{2}\right)^{2}}{24} - \frac{361}{96}$$
Trigonometric part
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2 7 5*x x - - - --- + -- 2 24 24
$$\frac{x^{2}}{24} - \frac{5 x}{24} - \frac{7}{2}$$
-7/2 - 5*x/24 + x^2/24
Assemble expression
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2 7 5*x x - - - --- + -- 2 24 24
$$\frac{x^{2}}{24} - \frac{5 x}{24} - \frac{7}{2}$$
-7/2 - 5*x/24 + x^2/24
Rational denominator
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2
-84 + x - 5*x
--------------
24
$$\frac{x^{2} - 5 x - 84}{24}$$
(-84 + x^2 - 5*x)/24
Combining rational expressions
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7 x*(-5 + x) - - + ---------- 2 24
$$\frac{x \left(x - 5\right)}{24} - \frac{7}{2}$$
-7/2 + x*(-5 + x)/24
Numerical answer
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-3.5 + 0.0416666666666667*x^2 - 0.208333333333333*x
-3.5 + 0.0416666666666667*x^2 - 0.208333333333333*x
Combinatorics
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(-12 + x)*(7 + x)
-----------------
24
$$\frac{\left(x - 12\right) \left(x + 7\right)}{24}$$
(-12 + x)*(7 + x)/24
Powers
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2 7 5*x x - - - --- + -- 2 24 24
$$\frac{x^{2}}{24} - \frac{5 x}{24} - \frac{7}{2}$$
-7/2 - 5*x/24 + x^2/24
Common denominator
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2 7 5*x x - - - --- + -- 2 24 24
$$\frac{x^{2}}{24} - \frac{5 x}{24} - \frac{7}{2}$$
-7/2 - 5*x/24 + x^2/24