Let k be the largest integer such that the equation $(x - 1)^2 + 2kx + 11 = 0$ has no real roots. If y is a positive real number, then the least possible value of $k/4y + 9y$ is

We need to find the largest integer k for which a quadratic has no real roots, then use that k to minimize an expression.

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Starting with: $(x-1)^2 + 2kx + 11 = 0$

$(x-1)^2 = x^2 - 2x + 1$

So our equation becomes:

$x^2 - 2x + 1 + 2kx + 11 = 0$

Combining like terms:

$x^2 + x(2k - 2) + 12 = 0$

We need the quadratic in standard form $ax^2 + bx + c = 0$ to apply the discriminant condition.

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For a quadratic $ax^2 + bx + c = 0$ to have no real roots, its discriminant must be negative:

$\Delta = b^2 - 4ac < 0$

In our case: $a = 1$, $b = (2k-2)$, $c = 12$

So: $(2k-2)^2 - 4(1)(12) < 0$

$(2k-2)^2 = 4k^2 - 8k + 4$

Therefore: $4k^2 - 8k + 4 - 48 < 0$

$4k^2 - 8k - 44 < 0$

Dividing by 4: $k^2 - 2k - 11 < 0$

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We need to solve $k^2 - 2k - 11 < 0$

Complete the square:

$k^2 - 2k - 11 = (k^2 - 2k + 1) - 1 - 11 = (k-1)^2 - 12$

So: $(k-1)^2 - 12 < 0$

$(k-1)^2 < 12$

Taking square roots: $|k-1| < \sqrt{12} = 2\sqrt{3} \approx 3.46$

This gives us: $-2\sqrt{3} < k-1 < 2\sqrt{3}$

$1-2\sqrt{3} < k < 1+2\sqrt{3}$

$-2.46 < k < 4.46$

Since we want the largest integer k, we get $k = 4$.

Quick check: Does $k = 4$ work? $(4-1)^2 = 9 < 12$

Does $k = 5$ work? $(5-1)^2 = 16 > 12$

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Now we need to minimize $\frac{k}{4y} + 9y$ where $k = 4$ and $y > 0$.

This becomes: $\frac{4}{4y} + 9y = \frac{1}{y} + 9y$

This is a perfect setup for the AM-GM inequality.

For positive numbers $a$ and $b$:

$\frac{a + b}{2} \geq \sqrt{ab}$

Therefore: $a + b \geq 2\sqrt{ab}$

Let us apply this with $a = \frac{1}{y}$ and $b = 9y$:

$\frac{1}{y} + 9y \geq 2\sqrt{\frac{1}{y} \cdot 9y} = 2\sqrt{9} = 2 \times 3 = 6$

Equality occurs when $a = b$, so:

$\frac{1}{y} = 9y$

$1 = 9y^2$

$y^2 = \frac{1}{9}$

$y = \frac{1}{3}$ (since $y > 0$)

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The least possible value of $\frac{k}{4y} + 9y$ is 6.

Key insight: No real roots corresponds to a negative discriminant, and AM-GM inequality is perfect for minimizing expressions of the form $\frac{a}{x} + bx$.