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secondary 3 | A Maths
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please help me with (ii)
Date Posted: 2 years ago
Views: 336
y = 2x² + (h + 4)x + 2h
Discriminant of roots of equation
= (h + 4)² - 4 (2) (2h)
= h² + 4h + 4h + 16 - 16h
= h² - 8h + 16
= (h - 4) (h - 4)
= (h - 4)²
Discriminant of roots of equation
= (h + 4)² - 4 (2) (2h)
= h² + 4h + 4h + 16 - 16h
= h² - 8h + 16
= (h - 4) (h - 4)
= (h - 4)²
Now, (h - 4)² is a squared expression as usual - it has a non-negative output.
h = 4 leads to a zero value, while all other values of h leads to a positive value.
So, our discriminant is either zero or positive ALL the time.
A positive discriminant means that our curve will cut the x-axis twice. Suffice to say, this requires that the curve enters the negative portion of y. Howsoever you try to draw a quadratic curve with two x-intercepts, the curve must exist on both the positive and negative regions.
That means, for y to "cannot be negative", our discriminant cannot be positive.
There is only one possibility for this to occur in this scenario.
If our discriminant (h - 4)² is zero or positive and we cannot have it positive...
...then the only way is that our discriminant if zero. This means that the curve will just "touch-and-go" at the x-axis, just like how you would touch-and-go during 4 x 10 m shuttle run,
So, (h - 4)² = 0 and therefore h = 4.
h = 4 leads to a zero value, while all other values of h leads to a positive value.
So, our discriminant is either zero or positive ALL the time.
A positive discriminant means that our curve will cut the x-axis twice. Suffice to say, this requires that the curve enters the negative portion of y. Howsoever you try to draw a quadratic curve with two x-intercepts, the curve must exist on both the positive and negative regions.
That means, for y to "cannot be negative", our discriminant cannot be positive.
There is only one possibility for this to occur in this scenario.
If our discriminant (h - 4)² is zero or positive and we cannot have it positive...
...then the only way is that our discriminant if zero. This means that the curve will just "touch-and-go" at the x-axis, just like how you would touch-and-go during 4 x 10 m shuttle run,
So, (h - 4)² = 0 and therefore h = 4.
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Hope you can understand better. You must understand what is the meaning of b2-4ac.
a positive means it is “u” shape curve. For this case a is positive, so you will have a “u” shape curve. For y to can be negative, b2-4ac must bigger than zero (you will have two real roots)
If it is equal to zero, (you will have one root, y is equal to zero and touching one point at x-axis)
For this case, b2-4ac is always bigger than zero, except h=4.
a positive means it is “u” shape curve. For this case a is positive, so you will have a “u” shape curve. For y to can be negative, b2-4ac must bigger than zero (you will have two real roots)
If it is equal to zero, (you will have one root, y is equal to zero and touching one point at x-axis)
For this case, b2-4ac is always bigger than zero, except h=4.
Date Posted:
2 years ago
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