Math / Algebra

QuestionP'education nationale Oujda Angad Lycée Salam Oujda En Mathématiques Exercice 1 : (2points) Factoriser les expressions suivantes : 1 pt $A=4 x^{2}-(x-1)^{2}$ 1 pt $B=x^{2}-27$

Studdy Solution

STEP 1

1. We are asked to factorize two algebraic expressions.
2. The expressions can be simplified using algebraic identities and factoring techniques.

STEP 2

1. Factorize the expression $A = 4x^2 - (x-1)^2$ .
2. Factorize the expression $B = x^2 - 27$ .

STEP 3

Expand the expression $(x-1)^2$ in $A = 4x^2 - (x-1)^2$ .
$(x-1)^2 = x^2 - 2x + 1$
Substitute back into the expression for $A$ :
$A = 4x^2 - (x^2 - 2x + 1)$

STEP 4

Simplify the expression by distributing the negative sign:
$A = 4x^2 - x^2 + 2x - 1$
Combine like terms:
$A = 3x^2 + 2x - 1$

STEP 5

Factor the quadratic expression $3x^2 + 2x - 1$ .
To factor, find two numbers that multiply to $3 \times (-1) = -3$ and add to $2$ . These numbers are $3$ and $-1$ .
Rewrite the middle term using these numbers:
$3x^2 + 3x - x - 1$
Group terms:
$(3x^2 + 3x) + (-x - 1)$
Factor by grouping:
$3x(x + 1) - 1(x + 1)$
Factor out the common factor $(x + 1)$ :
$A = (3x - 1)(x + 1)$

STEP 6

Factor the expression $B = x^2 - 27$ .
Recognize this as a difference of squares:
$B = x^2 - (\sqrt{27})^2$
Use the difference of squares formula $a^2 - b^2 = (a - b)(a + b)$ :
$B = (x - \sqrt{27})(x + \sqrt{27})$
Simplify $\sqrt{27}$ to $3\sqrt{3}$ :
$B = (x - 3\sqrt{3})(x + 3\sqrt{3})$

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QuestionP'education nationale Oujda Angad Lycée Salam Oujda En Mathématiques Exercice 1 : (2points) Factoriser les expressions suivantes : 1 pt $A=4 x^{2}-(x-1)^{2}$ 1 pt $B=x^{2}-27$

Studdy Solution

STEP 1

1. We are asked to factorize two algebraic expressions.
2. The expressions can be simplified using algebraic identities and factoring techniques.

STEP 2

1. Factorize the expression $A = 4x^2 - (x-1)^2$ .
2. Factorize the expression $B = x^2 - 27$ .

STEP 3

Expand the expression $(x-1)^2$ in $A = 4x^2 - (x-1)^2$ .
$(x-1)^2 = x^2 - 2x + 1$
Substitute back into the expression for $A$ :
$A = 4x^2 - (x^2 - 2x + 1)$

STEP 4

Simplify the expression by distributing the negative sign:
$A = 4x^2 - x^2 + 2x - 1$
Combine like terms:
$A = 3x^2 + 2x - 1$

STEP 5

STEP 6

Factor the expression $B = x^2 - 27$ .
Recognize this as a difference of squares:
$B = x^2 - (\sqrt{27})^2$
Use the difference of squares formula $a^2 - b^2 = (a - b)(a + b)$ :
$B = (x - \sqrt{27})(x + \sqrt{27})$
Simplify $\sqrt{27}$ to $3\sqrt{3}$ :
$B = (x - 3\sqrt{3})(x + 3\sqrt{3})$

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QuestionP'education nationale Oujda Angad Lycée Salam Oujda En Mathématiques Exercice 1 : (2points) Factoriser les expressions suivantes : 1 pt A=4x2−(x−1)2A=4 x^{2}-(x-1)^{2}A=4x2−(x−1)2 1 pt B=x2−27B=x^{2}-27B=x2−27

Studdy Solution

STEP 1

1. We are asked to factorize two algebraic expressions.2. The expressions can be simplified using algebraic identities and factoring techniques.

STEP 2

1. Factorize the expression A=4x2−(x−1)2 A = 4x^2 - (x-1)^2 A=4x2−(x−1)2.2. Factorize the expression B=x2−27 B = x^2 - 27 B=x2−27.

STEP 3

Expand the expression (x−1)2 (x-1)^2 (x−1)2 in A=4x2−(x−1)2 A = 4x^2 - (x-1)^2 A=4x2−(x−1)2.(x−1)2=x2−2x+1 (x-1)^2 = x^2 - 2x + 1 (x−1)2=x2−2x+1Substitute back into the expression for A A A:A=4x2−(x2−2x+1) A = 4x^2 - (x^2 - 2x + 1) A=4x2−(x2−2x+1)

STEP 4

Simplify the expression by distributing the negative sign:A=4x2−x2+2x−1 A = 4x^2 - x^2 + 2x - 1 A=4x2−x2+2x−1Combine like terms:A=3x2+2x−1 A = 3x^2 + 2x - 1 A=3x2+2x−1

STEP 5

STEP 6

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QuestionP'education nationale Oujda Angad Lycée Salam Oujda En Mathématiques Exercice 1 : (2points) Factoriser les expressions suivantes : 1 pt $A=4 x^{2}-(x-1)^{2}$ 1 pt $B=x^{2}-27$

1. We are asked to factorize two algebraic expressions.
2. The expressions can be simplified using algebraic identities and factoring techniques.

1. Factorize the expression $A = 4x^2 - (x-1)^2$ .
2. Factorize the expression $B = x^2 - 27$ .

Expand the expression $(x-1)^2$ in $A = 4x^2 - (x-1)^2$ .
$(x-1)^2 = x^2 - 2x + 1$
Substitute back into the expression for $A$ :
$A = 4x^2 - (x^2 - 2x + 1)$

Simplify the expression by distributing the negative sign:
$A = 4x^2 - x^2 + 2x - 1$
Combine like terms:
$A = 3x^2 + 2x - 1$