Help!!! Wjec as sam maths unit 1 question 9

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Raaheel
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9) The quadratic equation 4x^2 􏰅- 12x􏰈 + m =􏰆0, where m is a positive constant, has two distinct real roots.
Show that the quadratic equation3x^2 􏰈+mx +􏰈7􏰆= 0 has no real roots.

I know how to the first quadratic equation but when I looked at the answer for second quadratic equation.

b2 –4ac=m^2 –84
m2 <81 ---> 􏰂b2 –4ac<–3
b2 –4ac<0􏰂 ---> no real roots
Where did 81 come from???
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Prasiortle
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(Original post by Raaheel)
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9) The quadratic equation 4x^2 􏰅- 12x􏰈 + m =􏰆0, where m is a positive constant, has two distinct real roots.
Show that the quadratic equation3x^2 􏰈+mx +􏰈7􏰆= 0 has no real roots.

I know how to the first quadratic equation but when I looked at the answer for second quadratic equation.

b2 –4ac=m^2 –84
m2 <81 ---> 􏰂b2 –4ac<–3
b2 –4ac<0􏰂 ---> no real roots
Where did 81 come from???
Since the first quadratic has two distinct real roots, we have (-12)^2 > 4(4)(m), i.e. m < 9. Since m is given to be positive, and 9 is certainly positive, we know that both sides of this inequality are positive, so we can safely square both sides to give m^2 < 81 (note that e.g. if one side is negative, then we can't necessarily square, e.g. -5 < 3 is true, but then squaring both sides gives 25 < 9 which is false. That is why we had to make sure that both sides were positive before squaring.)
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Raaheel
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(Original post by Prasiortle)
Since the first quadratic has two distinct real roots, we have (-12)^2 > 4(4)(m), i.e. m < 9. Since m is given to be positive, and 9 is certainly positive, we know that both sides of this inequality are positive, so we can safely square both sides to give m^2 < 81 (note that e.g. if one side is negative, then we can't necessarily square, e.g. -5 < 3 is true, but then squaring both sides gives 25 < 9 which is false. That is why we had to make sure that both sides were positive before squaring.)
Thank you so much
Then you use same value for m for the second quadratic equation and why?. I thought m is different in each case
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Prasiortle
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(Original post by Raaheel)
Thank you so much
Then you use same value for m for the second quadratic equation and why?. I thought m is different in each case
If they were different, the problem author would have used different letters, as otherwise the problem would be ambiguous. When you see a question that uses the same variable multiple times, you can be certain that they are referring to the same thing.
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Raaheel
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(Original post by Prasiortle)
If they were different, the problem author would have used different letters, as otherwise the problem would be ambiguous. When you see a question that uses the same variable multiple times, you can be certain that they are referring to the same thing.
Thank you so much, I was trying to figure that question for ages. Thank you
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