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October 27th, 2016, 09:43 PM   #1
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Quadratics

Hey, I was just wondering about this math problem. Given the parabola y=ax^2+bx+9, find the values of a and b if the parabola passes through the point (8,1) and ax^2+bx+9=0 has two equal roots. If anyone could solve this problem it would be great!
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October 27th, 2016, 10:00 PM   #2
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$y=ax^2 + b x + 9$

$(8,1 ) \Rightarrow 1 = a(8 )^2 + b(8 ) + 9 = 64a + 8b +9$

$-8 = 64a + 8b$

$-1 = 8a + b$

2 equal roots $\Rightarrow (b^2 - 4(a)(9))=0$

$b^2 = 36a$

I leave it to you to solve these two equations for $a,~b$
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October 28th, 2016, 01:46 AM   #3
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It's impossible for a quadratic to have two equal roots. When its discriminant is zero, a quadratic has just one root, sometimes called a repeated root or a root of multiplicity 2 (because the quadratic's factorization contains a linear factor that is repeated or raised to the power of 2).
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October 28th, 2016, 02:08 AM   #4
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Quote:
Originally Posted by skipjack View Post
> > > It's impossible for a quadratic to have two equal roots. < < <When its discriminant is zero, a quadratic has just one root, sometimes called a repeated root or a root of multiplicity 2 (because the quadratic's factorization contains a linear factor that is repeated or raised to the power of 2).
You've just described the same thing in the rest of your paragraph that you said doesn't exist in your
first sentence in different words. The two roots are one and the same. The roots are equal to each
other. The roots coincide with one another. Etc. They're just different ways of looking at it/describing it.

Last edited by skipjack; October 28th, 2016 at 03:10 AM.
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October 28th, 2016, 03:40 AM   #5
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If the discriminant is zero, there is just one root, not two roots that coincide or are the same. The phrases "repeated root" and "root of multiplicity 2" are not ideally worded, but would seem lengthy and clumsy if changed to "root corresponding to a repeated linear factor" and "root corresponding to a squared linear factor".
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October 28th, 2016, 01:38 PM   #6
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Quote:
Originally Posted by skipjack View Post
If the discriminant is zero, there is just one root, not two roots that coincide or are the same. The phrases "repeated root" and "root of multiplicity 2" are not ideally worded, but would seem lengthy and clumsy if changed to "root corresponding to a repeated linear factor" and "root corresponding to a squared linear factor".
No, you still aren't stating anything different. "Two roots that coincide or are
the same [root]" mean it's a repeated root.

"Two roots that coincide" mean that they have become one [root]. You have difficulty (or are stubborn)
with the English to mathematics, in my opinion.
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October 28th, 2016, 05:08 PM   #7
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Please agree to disagree. You are arguing semantics here.

-Dan
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October 29th, 2016, 03:04 PM   #8
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I'm going to agree with @skipjack. Love his impartiality and ruthlessness. He's sent me a few brutal replies, but they've kept my feet on the ground. I'm not trying to brown nose here, I'm simply stating an opinion. There's nothing wrong with dealing with semantics. They can help us clarify our thoughts.

Mind you, I did find this on the internet:

b2 - 4ac = 0 can someone briefly explain thanks - The Student Room

You may like to read it.
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