Question

${x}^{2} - 24x + 72$
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Answer 1

$\sf\huge\bold{Solution}$

We use quadratic formula to find out the roots of the given Equation :

x²-24x+72

Here,a=1 ,b=-24 and c =72

$\sf \: \boxed{ x = \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac } }{2a} }$

$\sf \longmapsto \: x = \dfrac{ - \bigg( - 24\bigg) \pm \sqrt{ {\bigg( - 24\bigg)}^{2} - 4 \times 72} }{2}$

$\sf \longmapsto \: x = \dfrac{ - \bigg( - 24\bigg) \pm \: \sqrt{576 - 4 \times 72} }{2}$

$\sf \longmapsto \: x = \dfrac{ - \bigg( - 24\bigg) \pm \sqrt{576 - 288} }{2}$

$\sf \longmapsto \: x = \dfrac{ - \bigg( - 24\bigg) \pm \sqrt{288} }{2}$

$\sf \longmapsto \: x = \dfrac{ - \bigg( - 24\bigg) \pm12 \sqrt{2} }{2}$

We have plus minus sign so it have two values one of plus and another of minus.

$\sf \longmapsto \: x = \dfrac{24 \pm12 \sqrt{2} }{2}$

$\sf \longmapsto \: x = \dfrac{2\bigg(12 + 6 \sqrt{2}\bigg) }{2}$

$\sf \: x = 12 + 6 \sqrt{2}$

$\sf \longmapsto \: x = \dfrac{24 - 12 \sqrt{2} }{2}$

$\sf \longmapsto \: x = \dfrac{2\bigg(12 - 6 \sqrt{2}\bigg) }{2}$

$\sf \: x = 12 - 6 \sqrt{2}$

Now ,the final answer is

$\sf\boxed{\bigg(x -\bigg (6 \sqrt{2} + 12\bigg)\bigg)\bigg(x - \bigg(12 - 6 \sqrt{2} \bigg)\bigg)}$

Answer 2

x²-24x+72

$\huge{\underline{\red{\mathfrak{♡ANSWER♡}}}}$

Given: 2x 2−24x+72

=2(x2-12x+36), (take 2 common) =2(x2 −6x−6x+36), (split middle term so that resultant

is same)

=2[(x2 −6x)+(−6x+36)],( group pair of terms)

=2[x(x−6)−6(x−6)], (factor each binomials)

=2(x−6)(x−6), (factor out common factor (x−6))

=2(x−6)²

Aliter, use (a+b 2)=a²−2ab+b²

⇒2(x3 −12x+36)=2(x2 −2×6x+6²)=2(x−6)²