Quadratic Equations Help

[jwlallen]

Hi all,

 

This is my first post & I hope I've put it in the right place. I'm a mature student retaking secondary school Maths by distance learning.

 

I'm currently on Quadratic Equations and I've got myself in a right pickle!

 

I'm ok with the theory but when it comes to applying it to word problems I'm struggling. Specifically the following problem which I've been banging my head against a wall to solve but just can't get:

 

"3 consecutive numbers are written as x, x+1, x+2. The square of the largest number is 45 less than the sum of the squares of the other numbers. Find the 3 numbers?"

 

I know the answer is 8, 9, 10 but I can't see how the solution is reached.

 

My distance learning course is pretty poor and no explanation is given how to solve this one. I've muddled my way through a few others but this one has me totally stumped.

 

Thank you in advance for any support you can give.

[studiot]

OK, here's the griff.

 

The variable x is what is called a 'parameter'.

 

The idea of a parameter is that we put everything in terms of that parameter.

 

That is we write equations connecting the parameter to the unknowns.

 

Since there is then one equation in only one unknown (the parameter) it can be solved.

 

You will find that parameters are used in many situations in maths to simplify and connect apparently unconnected quantities.

 

In your specific question there are three unknowns.

 

Let as call them A, B and C

 

So

A = x

B = (x+1)

C = (x+2)

 

Which is the largest?

 

Can you now write an equation connecting A, B and C (but not containing x) from the information given.

 

Post your equation or explain your difficulty

 

Can you see the next step?

Edited September 2, 2013 by studiot

[jwlallen]

You know what reading that I think it's just clicked a little.

 

I'm thinking:

 

 

x^2 + (x+1)^2 = (x+2)^2 + 45

x^2 + (x+1)^2 - (x+2)^2 = 45

x^2 + x^2 + x + x + 2 - x^2 + 2x + 4 = 45

X^2 + 4x + 6 = 45

x^2 + 4x + 6 - 45 = 0

x^2 + 4x - 39 = 0

 

I'm still going wrong somewhere but can't see it!

[Fuzzwood]

x^2 + x^2 + x + x + 2 - x^2 + 2x + 4 = 45 <-- this line is wrong. Do you know why? Furthermore 8,9,10 is not the only solution to this question. You will find out why later.

Edited September 2, 2013 by Fuzzwood

[studiot]

Yes it's your arithmetic.

 

Tell me, why did you not write the equation in the order that you presented the information in your post#1?

 

I did ask which was the largest.

 

The square of the largest (C²) equals the sum of the squares of the other two (A² +B²) minus 45

 

C² = (A² +B²) - 45

 

Substitute from my 3 equations for x

 

(x+2)² = x² + (x+1)² - 45

 

expand and collect terms

 

x² - 2x -48 = 0

 

note in these forums the superscript and subscript icons on the toolbar at the top of the edit window

[ewmon]

"3 consecutive numbers are written as x, x+1, x+2. The square of the largest number is 45 less than the sum of the squares of the other numbers. Find the 3 numbers?"

 

The square of the largest number

is

the sum of the squares of the other numbers

45 less than

(x+2)²

=

x² + (x+1)²

– 45

(x+2)² = x² + (x+1)² – 45

[jwlallen]

Hey all,

 

Just wanted to thank you all for your help with this. After tearing my hair out again yesterday and using your feedback i'm finally getting somewhere. Seems like screwing up on the more basic side of things was tripping me up (feels like every time i learn something slightly more advanced it pushes out something basic This alongside setting out the problem in a not so helpful way just made it even more complicated.

 

This is where i am so far:

 

"3 consecutive positive numbers are written as x, x+1, x+2. The square of the largest number is 45 less than the sum of the squares of the other numbers. Find the 3 numbers?"

Soooo....

 

(x²) + (x+1)² = (x+2)² + 45

x² + (x+1)(x+1) = (x+2)(x+2) + 45

x² + x² + x + x +1 = x² + 2x + 2x + 4 + 45

2x² + 2x + 1 = x² + 4x + 49

x² - 2x - 48 = 0

 

x = 8, -6

 

Since the question asks for positive consecutive numbers (apologies, I missed that in my orginal post) then I get:

 

x=8,

x+1=9,

x+2=10

 

So satisfying to nail this one. Thanks everyone that contributed. Just had a flick through what comes next and i'll be attempting quadratic fractions then simultaneous equations with quadratic fractions, so I'm sure I'll be back again soon.