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Chapter 17: Problem 534

If the equation \(\mathrm{x}^{2}+2(\mathrm{k}+2) \mathrm{x}+9 \mathrm{k}=0\) has equal roots, find \(\mathrm{k}\).

Short Answer

Expert verified
The equation \(x^2 + 2(k + 2)x + 9k = 0\) has equal roots when \(k = 1\) or \(k = 4\).

Step by step solution

01

Write down the discriminant formula and input the given values

The discriminant formula is: \(\Delta = b^2 - 4ac\). We are given \(a = 1\), \(b = 2(k + 2)\) and \(c = 9k\). Plug in the values in the discriminant formula: \[\Delta = [2(k + 2)]^2 - 4(1)(9k)\]
02

Simplify the discriminant equation

Now, we need to simplify the equation: \(\Delta = [2k+4]^2 -36k\) Expanding the equation, we get: \[\Delta = {(2k + 4)(2k + 4)} - 36k\] \[\Delta = (4k^2 + 16k + 16) - 36k\]
03

Set the discriminant to zero for equal roots

Since we are looking for equal roots, we need to set the discriminant equal to zero: \[\Delta = 4k^2 + 16k + 16 - 36k = 0\]
04

Solve for k

Now we need to solve for k: \[4k^2 - 20k + 16 = 0\] Divide through by 4 to simplify the equation: \[k^2 - 5k + 4 = 0\] Now we have a quadratic equation which can be factored: \[(k - 1)(k - 4) = 0\] This gives us two possible values for k: \[k = 1\] or \[k = 4\] Hence, the equation \(x^2 + 2(k + 2)x + 9k = 0\) has equal roots when \(\textbf{k = 1}\) or \(\textbf{k = 4}\).

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