This came in my exam today

Q) Write the expression 2x26x+15 -2x^2 - 6x + 15 in the form of a(x+p)2+q a ( x + p)^2 + q where a, p and q are integers.

Can anyone help me out.?

Please send your answers as complete solutions. Thank you

Note by Syed Hamza Khalid
1 year, 4 months ago

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1 vote

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Here is my full solution :

a(x+p)2+q=2x26x+15a(x2+2px+p2)+q=2x26x+15ax2+2apx+ap2+q=2x26x+15\begin{aligned} a(x + p)^2 + q & = - 2x^2 - 6x + 15 \\ a(x^2 + 2px + p^2) + q & = -2x^2 - 6x + 15 \\ {\color{#3D99F6}a}x^2 + {\color{#E81990}2ap}x + {\color{#20A900} ap^2 + q} & = {\color{#3D99F6}-2}x^2 {\color{#E81990}-6}x + {\color{#20A900}15} \\ \end{aligned}

On comparing both the sides we get, a=2,p=32,q=392a = -2, p = \dfrac32, q = \dfrac{39}{2}. So, the final result is

2x26x+15=2(x32)2+392-2x^2 - 6x + 15 = -2 \left( x - \dfrac32 \right)^2 + \dfrac{39}{2}


@Syed Hamza Khalid, as you said there is no possibility for p,qp,q to be integers. So, the question must be having a typo or it wants to say rational numbers instead of integers.

Ram Mohith - 1 year, 4 months ago

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Yeah thats wat I think

Syed Hamza Khalid - 1 year, 4 months ago

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Even I don't know- I got 2(x+32)2212 -2( x + \dfrac{3}{2} )^2 - \dfrac{21}{2} but the question asked a,pa, p and q q are integers.. and these are not \therefore I think there is a typo error in the question perhaps. But how did u get +392 + \dfrac{39}{2} at the end

Syed Hamza Khalid - 1 year, 4 months ago

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ap2+q=15-ap^2 + q = 15 from which we will get q.

Ram Mohith - 1 year, 4 months ago

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I just used complete the square: =2x26x+15=2(x2+3x)+15=2((x+32)294)15=2(x+32)2+9215=2(x+32)2212. = -2x^2 - 6x +15 \\ = -2(x^2 + 3x) + 15 \\ = -2((x + \dfrac{3}{2})^2 - \dfrac{9}{4} ) - 15 \\ = -2(x + \dfrac{3}{2})^2 + \dfrac{9}{2} - 15 \\ = -2(x + \dfrac{3}{2} )^2 - \dfrac{21}{2} . \square


This was the method we learned at school \therefore I applied it but I don't understand why I didn't get integers

Syed Hamza Khalid - 1 year, 4 months ago

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@Syed Hamza Khalid I will check once again.

Ram Mohith - 1 year, 4 months ago

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@Syed Hamza Khalid Why did you write "-15" in the third step instead of "+15" ?

Ram Mohith - 1 year, 4 months ago

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I got it as 2(x+32)2+392-2(x + \dfrac32)^2 + \dfrac{39}{2}. Is it correct.

Ram Mohith - 1 year, 4 months ago

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Hah, what a question! Have you used a research paper service like https://ca.edubirdie.com/research-paper-writing-services to complete these tasks or not? I'm gonna pass my exams soon and want to know the answer.

Eveline Heston - 6 months, 2 weeks ago

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