Analysis of errors in formulas

This page shows the results of analysis of error distribution over formulas used within four performed tasks. You see the total number of errors per particular formula. Observe that the main source of errors is commutative law; accordingly, the recommended exercise focuses on application of this law.

Analysis: formulas - All performed tasks

Contents

	Table
	Plot
	Methodical recommendations:

Methodical recommendations:

Perform the task

Rational equations

Fractional algebraic equations

Fractional algebraic equations reduced to linear equations

Solve the equation

\frac{3 t + 4}{2 t - 3} + \frac{4 t - 3}{- 4 t - 3} = \frac{1}{2}

Solve the problem with any of the following techniques

	Scheme "To view 4"
	Scheme "To relate the step with category 3"
	Scheme "To learn categories 3"
	Scheme "To view and learn categories 3"
	Scheme "To relate categories with the step 3"
	Scheme "To select for transformation 2"

Table

N	Formula	Errors:	Hints:	Total:
0	$a + b = b + a$	4	1	5
1	$P + a = Q <=> P = Q - a$	3	0	3
2	$a \cdot (b + c) = a \cdot b + a \cdot c$	2	0	2
3	$a + b = c$	1	1	2
4	$x^{2} + b x + c = t$	2	0	2
5	$f (x) = 0 \exists Ω \subset R : {\begin{cases} \forall λ \in Ω f (λ) \equiv 0 \\ \forall λ \notin Ω f (λ) \neq 0 \end{cases} \Rightarrow x \in Ω$	0	2	2
6	$k x + b = 0 <=> x = - k^{-1} b$	0	2	2
7	$a x^{2} + b x + c = a \cdot (x - \frac{- b + \sqrt[2]{b^{2} - 4 a c}}{2 a}) \cdot (x - \frac{- b - \sqrt[2]{b^{2} - 4 a c}}{2 a})$	2	0	2
8	$\prod_{k = 0}^{n} f_{k} (x) = 0 \Rightarrow [\begin{array}{l} f_{k} (x) = 0 \\ k = 0,1,... n \end{array}$	0	1	1
9	$x = f (t) => t = f^{-1} (x)$	0	1	1
10	${(a \pm b)}^{2} = a^{2} \pm 2 a b + b^{2}$	0	1	1
11	$f (x) \geq a x \in Ω \cap D (f)$	1	0	1
12	$\begin{array}{l} {(a + b)}^{n} = \sum_{k = 0}^{n} C_{n}^{k} \cdot a^{(n - k)} \cdot b^{k}, \\ {\begin{cases} C_{n}^{k} = \frac{n!}{(n - k)! \cdot k!} \\ 0! = 1 n! = (n - 1)! \cdot n \end{cases} \\ E x a m p l e 1 : {(a \pm b)}^{5} = C_{5}^{0} \cdot a^{5} \pm C_{5}^{1} \cdot a^{4} \cdot b + C_{5}^{2} \cdot a^{3} \cdot b^{2} \pm C_{5}^{3} \cdot a^{2} \cdot b^{3} + C_{5}^{4} \cdot a \cdot b^{4} \pm C_{5}^{5} \cdot b^{5} \\ {(a \pm b)}^{5} = a^{5} \pm 5 \cdot a^{4} \cdot b + 10 \cdot a^{3} \cdot b^{2} \pm 10 \cdot a^{2} \cdot b^{3} + 5 \cdot a \cdot b^{4} \pm b^{5} \\ E x a m p l e 2 : C_{7}^{3} = \frac{7!}{(7 - 3)! \cdot 3!} = \frac{4! \cdot 5 \cdot 6 \cdot 7}{4! \cdot 3!} = \frac{5 \cdot 6 \cdot 7}{1 \cdot 2 \cdot 3} = 35 \end{array}$	0	1	1

Plot

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