Worksheet · solutions includedQuadratics — Practice Worksheet
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Section A — the questions you missed
Original — Math M2 Q13 (you answered 0)
In the equation x2 − 34x + c = 0, c is a constant. The equation has no real solutions if c > n. What is the least possible value of n?
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No real solutions → discriminant < 0: (−34)2 − 4(1)(c) < 0.
1156 − 4c < 0 → c > 289.
So the least value of n is 289.
Answer: 289
Original — Math M2 Q19 (you answered B; correct A)
57x2 + (57b + a)x + ab = 0, where a and b are positive constants. The product of the solutions is k·ab. What is k?
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By Vieta’s, product of roots = c/a = ab / 57.
That equals k·ab, so k·ab = ab/57 → k = 1/57.
Answer: A) 1/57
Original — Math M2 Q20 (omitted)
The graph of x2 + x + y2 + y = 199/2 is a circle. What is the radius?
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Complete the square in x and y: add (½)2 = ¼ for each.
(x + ½)2 + (y + ½)2 = 199/2 + ¼ + ¼ = 199/2 + ½ = 100.
r2 = 100 → r = 10.
Answer: 10
Section B — new practice (same skills)
Practice 1
x2 + 12x + c = 0 has no real solutions if c > n. Find the least n.
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D < 0: 122 − 4c < 0 → 144 < 4c → c > 36.
Answer: 36
Practice 2
The system y = 5 and y = x2 − 6x + a has exactly one real solution. What is a?
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x2 − 6x + a = 5 → x2 − 6x + (a − 5) = 0.
D = 0: (−6)2 − 4(a − 5) = 0 → 36 = 4(a − 5) → a − 5 = 9 → a = 14.
Answer: 14
Practice 3
The graph of x2 − 6x + y2 + 8y = 0 is a circle. Find its radius.
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Complete the square: (x − 3)2 − 9 + (y + 4)2 − 16 = 0.
(x − 3)2 + (y + 4)2 = 25 → r = 5.
Answer: 5
Practice 4
For 3x2 + 12x + 7 = 0, what is the sum of the solutions?
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Vieta’s: sum = −b/a = −12/3 = −4.
Answer: −4
Practice 5 (build from a table)
A quadratic f has f(−1)=10, f(0)=14, f(1)=20. Find f(x) = ax2+bx+c.
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f(0)=14 → c = 14.
f(1)=20 → a + b + 14 = 20 → a + b = 6.
f(−1)=10 → a − b + 14 = 10 → a − b = −4.
Add: 2a = 2 → a = 1, so b = 5.
f(x) = x2 + 5x + 14. (This mirrors M2 Q5.)
Answer: f(x) = x2 + 5x + 14