Quadratic Equations

1.
x² - 12x + 36 = 0
Determinant Δ
= 12² - 4*1*36
= 0


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2.
i)
5x² - 3 = 2x
5x² - 2x - 3= 0
Determinant Δ
= (-2)² - 4*5*(-3)
= 64
Since Δ > 0, there are two distinct real roots.

ii)
2(x² - 3x) = x - 7
2x² - 6x = x - 7
2x² - 7x + 7 = 0
Determinant Δ
= (-7)² - 4*2*7
= -7
Since Δ < 0, there are no real roots.


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3.
i)
Since x² - 8x + k =0has two distinct real roots, determinant Δ > 0
(-8)² -4*1*k > 0
64 - 4k > 0
4k < 64
k < 16

Since 3x² - 6x + (2 - k) =0 has two distinct real roots, determinant Δ > 0
6² - 4*3*(2 - k) > 0
36 - 24 + 12k > 0
12k > -12
k > -1


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4.
x² + 2x + m = 3(2x +1)
x² + 2x + m = 6x + 3
x² - 4x + (m -3) = 0

Since x² - 4x + (m -3) = 0has two distinct real roots, determinant > 0
(-4)² - 4*1*(m - 3) > 0
16 - 4m + 12 > 0
4m < 28
m < 7


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5.
a.
Since (2k + 1)x² + 5x + 1 = 0 has real roots, Δ ≥ 0
5² - 4*(2k + 1)*1 ≥ 0
25 - 8k - 4 ≥ 0
8k ≤ 21
k ≤ 21/8

b.
When k is maximum, i.e. k = 21/8
[2(21/8) + 1)x² +5x +1 = 0
(25/4)x² + 5x + 1 = 0
25x² + 20x + 4 = 0
(5x + 2)² = 0
x = -2/5 (double roots)


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6.
Since (k - 2)x² + 2kx + (k + 6) = 0 has no real roots, determinant < 0
(2k)² - 4(k - 2)(k + 6) < 0
4k² - 4(k² +4k - 12) < 0
4k² - 4k² - 16k+ 48 < 0
16k > 48
k > 3
The minimum value of k = 4