Literal Calculation - Revision history

Robert.adlington: /* Inequalities */

2025-12-06T11:14:11Z

Inequalities

← Older revision		Revision as of 11:14, 6 December 2025
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	* If a > 0:		* If a > 0:
	** Algebraic resolution. The solutions are the reals x such that <math ~~display="inline"~~>x \geq -\frac{b}{a}</math>		** Algebraic resolution. The solutions are the reals x such that <math>x \geq -\frac{b}{a}</math>
	** You write the set of solutions as: <math>S = [-\frac{b}{a}; +\infty[</math>		** You write the set of solutions as: <math>S = [-\frac{b}{a}; +\infty[</math>
	** You represent this set of the real number line as greater than <math>-\frac{b}{a}</math>		** You represent this set of the real number line as greater than <math>-\frac{b}{a}</math>

Robert.adlington: /* Inequalities */

2025-12-06T11:13:49Z

Inequalities

← Older revision		Revision as of 11:13, 6 December 2025
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	* If a > 0:		* If a > 0:
	** Algebraic resolution. The solutions are the reals x such that <math>x \geq -\frac{b}{a}</math>		** Algebraic resolution. The solutions are the reals x such that <math display="inline">x \geq -\frac{b}{a}</math>
	** You write the set of solutions as: <math>S = [-\frac{b}{a}; +\infty[</math>		** You write the set of solutions as: <math>S = [-\frac{b}{a}; +\infty[</math>
	** You represent this set of the real number line as greater than <math>-\frac{b}{a}</math>		** You represent this set of the real number line as greater than <math>-\frac{b}{a}</math>

Robert.adlington: /* Inequalities */

2025-12-06T11:13:24Z

Inequalities

← Older revision		Revision as of 11:13, 6 December 2025
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	When you multiply two members of an inequality by the same number:		When you multiply two members of an inequality by the same number:
	* You keep the sense of the inequality if the number is positive or null. If <math>a \leq b</math> and <math>c \geq 0</math>then <math>ac \leq bc</math>		* You keep the sense of the inequality if the number is positive or null. If <math>a \leq b</math> and <math>c \geq 0</math>then <math>ac \leq bc</math>
	* You change the sense of the inequality if the number is negative or null. If <math>a \leq b</math> and <math>c \leq 0</math>then <math>ac \geq bc</math>		* You change the sense of the inequality if the number is negative or null. If <math>a \leq b</math> and <math>c \leq 0</math> then <math>ac \geq bc</math>
	You can multiply, member by member the inequalities of the same sense when all the terms are positive or null:		You can multiply, member by member the inequalities of the same sense when all the terms are positive or null:

	* If <math>0 \leq a \leq b</math>and <math>0 \leq c \leq d</math>then <math> ac \leq bd</math>		* If <math>0 \leq a \leq b</math>and <math>0 \leq c \leq d</math> then <math> ac \leq bd</math>

	When resolving an inequality like <math>ax + b \geq 0</math>you look first at the sign of a and then resolve in three steps:		When resolving an inequality like <math>ax + b \geq 0</math> you look first at the sign of a and then resolve in three steps:

	* If a > 0:		* If a > 0:

Robert.adlington: /* Inequalities */

2025-12-06T11:12:56Z

Inequalities

← Older revision		Revision as of 11:12, 6 December 2025
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	For reals:		For reals:

	* If <math>a \leq b</math>then <math>a + c \leq b + c		* If <math>a \leq b</math> then <math>a + c \leq b + c
	</math>		</math>
	* If <math>a \leq b</math>and <math>c \leq d</math>then <math>a + c \leq b + c		* If <math>a \leq b</math> and <math>c \leq d</math> then <math>a + c \leq b + c
	</math>		</math>
	When you multiply two members of an inequality by the same number:		When you multiply two members of an inequality by the same number:

Robert.adlington: /* Inequalities */

2025-12-06T11:12:33Z

Inequalities

← Older revision		Revision as of 11:12, 6 December 2025
Line 46:		Line 46:
	</math>		</math>
	When you multiply two members of an inequality by the same number:		When you multiply two members of an inequality by the same number:
	* You keep the sense of the inequality if the number is positive or null. If <math>a \leq b</math>and <math>c \geq 0</math>then <math>ac \leq bc</math>		* You keep the sense of the inequality if the number is positive or null. If <math>a \leq b</math> and <math>c \geq 0</math>then <math>ac \leq bc</math>
	* You change the sense of the inequality if the number is negative or null. If <math>a \leq b</math>and <math>c \leq 0</math>then <math>ac \geq bc</math>		* You change the sense of the inequality if the number is negative or null. If <math>a \leq b</math> and <math>c \leq 0</math>then <math>ac \geq bc</math>
	You can multiply, member by member the inequalities of the same sense when all the terms are positive or null:		You can multiply, member by member the inequalities of the same sense when all the terms are positive or null:

Robert.adlington: /* Inequalities */

2025-12-06T11:12:10Z

Inequalities

@@ Line 43: / Line 43: @@
 * If <math>a \leq b</math>then <math>a + c \leq b + c
 </math>
-* If <math>a \leq b</math>and <math>c \leq d</math>then <math>ac \leq bc
+* If <math>a \leq b</math>and <math>c \leq d</math>then <math>a + c \leq b + c
 </math>
-* If <math>a \leq b</math>and <math>c \geq 0</math>then
+When you multiply two members of an inequality by the same number:

Robert.adlington: /* Square Roots */

2025-12-06T10:53:58Z

Square Roots

← Older revision		Revision as of 10:53, 6 December 2025
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	* Thanks to the third remarkable identity, <math>(a + b\sqrt{n})(a - b\sqrt{n}) = a^2 - nb^2</math>		* Thanks to the third remarkable identity, <math>(a + b\sqrt{n})(a - b\sqrt{n}) = a^2 - nb^2</math>
	* In an expression like <math>\frac{A}{a + b\sqrt{2}}</math>you can simplify the denominator: <math>\frac{A}{a + b\sqrt{2}} = \frac{A(a - b\sqrt{2})}{a^2 - 2b^2}</math>		* In an expression like <math>\frac{A}{a + b\sqrt{2}}</math>you can simplify the denominator: <math>\frac{A}{a + b\sqrt{2}} = \frac{A(a - b\sqrt{2})}{a^2 - 2b^2}</math>

			== Equations ==

			* If <math>ax + b = 0</math>, if a is non-null, <math>x = -\frac{b}{a}</math>
			* If A = 0 or B = 0, then AB = 0

			== Inequalities ==
			For reals:

			* If <math>a \leq b</math>then <math>a + c \leq b + c
			</math>
			* If <math>a \leq b</math>and <math>c \leq d</math>then <math>ac \leq bc
			</math>
			* If <math>a \leq b</math>and <math>c \geq 0</math>then

Robert.adlington: /* Powers and Exponents */

2025-12-06T10:42:56Z

Powers and Exponents

← Older revision		Revision as of 10:42, 6 December 2025
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	* When <math>n = 0 ; a^0 = 1</math>		* When <math>n = 0 ; a^0 = 1</math>
	* If n is a natural integer et a is a non-null real number: <math>a^{-n} = \frac{1}{a^n}</math>		* If n is a natural integer et a is a non-null real number: <math>a^{-n} = \frac{1}{a^n}</math>

			== Square Roots ==
			If a is a positive real number, then there are two reals (one positive and one negative) for which the square is a.

			* The positive one is written as:<math>\sqrt{a}</math>
			* If a = 0, we can write <math>\sqrt{0} = 0</math>
			* The formula <math>\sqrt{a^2} </math> is equal to a or -a
			* For two non-null positive reals: <math>\sqrt{ab} = \sqrt{a}\sqrt{b}</math>
			* If a is a positive real or null and b is positive: <math>\sqrt{\frac{a}{b}} =\frac{\sqrt{a}}{\sqrt{b}}</math>
			* In particular: <math>\sqrt{\frac{1}{b}} =\frac{1}{\sqrt{b}}</math>
			* In an expression like <math>\frac{A}{\sqrt{b}}</math>, you can make the denominator rational: <math>\frac{A}{\sqrt{b}} = \frac{A\sqrt{b}}{(\sqrt{b})^2} = \frac{A\sqrt{b}}{b}</math>
			* Thanks to the third remarkable identity, <math>(a + b\sqrt{n})(a - b\sqrt{n}) = a^2 - nb^2</math>
			* In an expression like <math>\frac{A}{a + b\sqrt{2}}</math>you can simplify the denominator: <math>\frac{A}{a + b\sqrt{2}} = \frac{A(a - b\sqrt{2})}{a^2 - 2b^2}</math>

Robert.adlington: Created page with "== Developing and Factorizing Equations == For any real numbers: * $a(b + c) = ab + ac$ * $a(b - c) = ab - ac$ * $(a + b)(c + d) = ac + ad + bc + bd$ == Remarkable Identities == For any real numbers: * Square of a sum: $(a + b)^2 = a^2 + 2ab + b^2$ * Square of a difference: $(a - b)^2 = a^2 - 2ab + b^2$ * Difference of two squares: $a^2 - b^2 = (a + b)(a - b)$ == Powers and Exponents == For any non-nul..."

2025-12-06T10:22:20Z

Created page with "== Developing and Factorizing Equations == For any real numbers: * <math>a(b + c) = ab + ac</math> * <math>a(b - c) = ab - ac</math> * <math>(a + b)(c + d) = ac + ad + bc + bd</math> == Remarkable Identities == For any real numbers: * Square of a sum: <math>(a + b)^2 = a^2 + 2ab + b^2</math> * Square of a difference: <math>(a - b)^2 = a^2 - 2ab + b^2</math> * Difference of two squares: <math>a^2 - b^2 = (a + b)(a - b)</math> == Powers and Exponents == For any non-nul..."

New page

== Developing and Factorizing Equations ==
For any real numbers:

* <math>a(b + c) = ab + ac</math>
* <math>a(b - c) = ab - ac</math>
* <math>(a + b)(c + d) = ac + ad + bc + bd</math>

== Remarkable Identities ==
For any real numbers:

* Square of a sum: <math>(a + b)^2 = a^2 + 2ab + b^2</math>
* Square of a difference: <math>(a - b)^2 = a^2 - 2ab + b^2</math>
* Difference of two squares: <math>a^2 - b^2 = (a + b)(a - b)</math>

== Powers and Exponents ==
For any non-null natural integer:

* <math>a^n = \underbrace{ a * a * a * \cdots * a }_{n}</math>
* When <math>n = 0 ; a^0 = 1</math>
* If n is a natural integer et a is a non-null real number: <math>a^{-n} = \frac{1}{a^n}</math>