Question

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: Factrization$
$1) \: 3 {x}^{2} - x - 9$
$2) \: 12 {x}^{2} - 7x + 1$
$3) {x}^{2} - 14x + 48$


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Answer 1

$\qquad \maltese \; {\underline{\purple{\underline{\pmb{\bf{Given : }}}}}}$

$\bullet \; {\underline{\boxed{\tt{ 3x^2 - x - 9 }}}}$

$\bullet \; {\underline{\boxed{\tt{ 12x^2 - 7x +1}}}}$

$\bullet \; {\underline{\boxed{\tt{ x^2 - 14x + 48 }}}}$

$\qquad \maltese \; {\underline{\purple{\underline{\pmb{\bf{To \; Find : }}}}}}$

★ The factorised from of the expressions given

$\qquad \maltese \; {\underline{\purple{\underline{\pmb{\bf{Full \; Solution: }}}}}}$

★ Now here wee can see that we have been provided with 3 polynomials which could be classified into a Quadratic trinomial , Since the expressions are quadratic [ with degree 2 ] that means we can factorise the expression into 2 suitable factors

➤ How to factorise ?

Now we know that the general form of a quadratic expression is such that

$\bigstar \; {\underline{\boxed{\tt{ ax^2 + bx + c }}}}$ where the value x is considered to be the zero and  a , b , c are the constant terms following the condition a ≠ 0

We basically factorise the expression using the method of splitting middle term into such forms that ;

• The product of the middle terms equals ax² × c
• The sum of the middle terms equals to bx

Now keeping the above laws under notice let's perform the required action

⋆ Question 1 :

$\bullet \; {\underline{\boxed{\tt{ 3x^2 - x - 9 }}}}$

⋆ RequirEd AnswEr :

${ : \implies } \bf 3x^2 - x - 9$

• Now since the is no such whole number which exists that could satisfy the condition so , let's use the quadratic formula to solve it

${ : \implies } \bf \dfrac{-b \pm \sqrt{b^2 - 4ac} }{2a}$

${ : \implies } \bf \dfrac{-1 \pm \sqrt{1^2 - 4(3)(9)} }{2(3)}$

${ : \implies } \bf \dfrac{-1 \pm \sqrt{1 + 108} }{6}$

${ : \implies } \bf \dfrac{-1 \pm \sqrt{1 09} }{6}$

• Since the delta ( Δ ) is  not a perfect square so , the expression can't bee factorised

⋆ Question 2 :

$\bullet \; {\underline{\boxed{\tt{ 12x^2 - 7x +1}}}}$

⋆ RequirEd AnswEr  :

${ : \implies } \bf 12x^2 - 7x + 1$

${ : \implies } \bf 12x^2 - 4x -3x + 1$

${ : \implies } \bf 4x(3 x - 1 ) - 1 ( 3x - 1 )$

${ : \implies } \bf ( 4x - 1 ) ( 3x - 1 )$

⋆ Question 3 :

$\bullet \; {\underline{\boxed{\tt{ x^2 - 14x + 48 }}}}$

⋆ RequirEd AnswEr :

${ : \implies } \bf x^2 - 14x + 48$

${ : \implies } \bf x^2 - 7x - 7x + 48$

${ : \implies } \bf x^2 - 7x - 7x + 48$

${ : \implies } \bf x( x - 7 ) - 7 ( x - 7 )$

${ : \implies } \bf ( x - 7 ) ( x - 7 )$

$\qquad \maltese \; {\underline{\purple{\underline{\pmb{\bf{Therefore : }}}}}}$

• The factors of the expression are such that

$\bigstar \; {\underline{\boxed{\tt{ ( 4x- 1 ) ( 3x - 1 )}}}}$

$\bigstar \; {\underline{\boxed{\tt{ ( x- 7 ) ( x - 7 )}}}}$

$\qquad \maltese \; {\underline{\purple{\underline{\pmb{\bf{More \; To \; Know : }}}}}}$

★ The general form of a polynomial is

$\rightarrow \bf a_0 + a_1x +a_2x^2 + ...+ a_{n-1} x^{n-1} + a_nx^n$

★ The nature of the roots are such that

• They are real and distinct when D > 0
• They are imaginary when D < 0
• They are equal when D = 0

★ Some algebric Identities :

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c} \\ \tiny\bf{\dag}\:\underline{\frak{\rm{S}\frak{ome\:important\:algebric\:identities\:::}}} \\\\ \green{\bigstar}\:\rm \red{ (A+B)^{2} = A^{2} + 2AB + B^{2}} \\\\ \red{\bigstar}\rm\: \green{(A-B)^{2} = A^{2} - 2AB + B^{2}} \\\\ \orange{\bigstar}\rm\: \blue{A^{2} - B^{2} = (A+B)(A-B)}\\\\ \blue{\bigstar}\rm\: \orange{(A+B)^{2} = (A-B)^{2} + 4AB}\\\\ \pink{\bigstar}\rm\: \purple{(A-B)^{2} = (A+B)^{2} - 4AB}\\\\ \purple{\bigstar} \rm\: \pink{(A+B)^{3} = A^{3} + 3AB(A+B) + B^{3}}\\\\ \gray{\bigstar}\rm\:(A-B)^{3} = A^{3} - 3AB(A-B) + B^{3}\\\\ \bigstar\rm\: \gray{A^{3} + B^{3} = (A+B)(A^{2} - AB + B^{2})} \\\\ \end{array}}\end{gathered}\end{gathered}$

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Answer 2

Answer:

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