Equalities and the quadratic formula.

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John Vezina

20 april 2020 17:51

I need help understanding why I am having such difficulty with a certain problem. I am to solve 10 - (9x^2) + (4x) = (-6x^2)

Combining like terms I get (-3x^2) + (4x) + 10 = 0

so that: a = -3, b = 4, and c = 10 to use in the quadratic formula

Using the quadratic formula, for the term under the square root I get the square root of 136.

The correct answer is the square root of 34.

My quadratic formula calculation is - 4 (+/-) Square root of (4^2) - 4(-3) (10) all divided by -6

The correct answer is: -2 (+/-) root 34 all divided by -3

I think that maybe the only source of a mistake I might have made is in the a, b, and c terms so I likely did something wrong when combining terms in the first place.

The second part of my question is why am I getting a different result if I move the left hand side of the above original equation to the right, making the left side zero, and another result if I move the right hand side to the left making the right side zero? The difference results in a reflected parabola, reflected across the x axis.

I'm hopeful someone can point out what I am doing wrong.

Thanks.

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John Vezina

20 april 2020 22:37
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Wow! Thank you so much. I'm a little rusty on breaking down the square roots, so I will google that process to get up to speed with it. I'm glad I did not get the quadratic formula wrong and I can solve it but got blindsided by the sq.root of 136 thing. Could you comment on the fact that subtracting the right side from both sides gives you 0 = 3x^2 - 4x - 10 which is the reflection of -3x^2 + 4x + 10 = 0

I find this odd, as intuitively I would have thought that "something" = 0 would be the same as 0 = "something".
John Vezina

21 april 2020 15:42
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Leo,

Again thank you lots for your help, it is very much appreciated. You are right and I now know I made a mistake in thinking the two parabolas were exact reflections. If they were exact reflections, the vertexs would be identical. I am also using geogebra app on my iPad and love the app. I’m downloading the PC version right now thanks to your heads up. I’m learning to pay more attention to the details rather than trying to rush through to an answer. I’m still thinking about the overall structure of the initial (or any) equation as one looks at it on paper. I’m not very talented mathematically, so I get confused easily. For example, I had assumed that because it is an equation, it did not matter what side was brought to zero by inverse operations, but now I suddenly realize that by choosing which side to have your zero on, you will get a different answer. And I know why as well. In the first case, you are adding -6x^2 from both sides. In the second case, you are subtracting (10 - 9x^2 - 4^x) from both sides.

Now I will be looking at equations in a very different light. I’m not sure if you understand my concern here, but there you have it.

Thanks,

John

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Equalities and the quadratic formula.

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