Multiple choice

Find the value of $k$ such that the quadratic equation $\displaystyle { x }^{ 2 }-\left( k+6 \right) x+2\left( 2k-1 \right) =0$ has sum of the roots as half of their product.

  1. 1

  2. 2

  3. 3

  4. 7
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D Correct answer
Explanation

For x^2 - (k+6)x + 2(2k-1) = 0, sum of roots = k+6 and product of roots = 2(2k-1). Given sum = 1/2 * product, so k+6 = 1/2 * 2(2k-1) = 2k-1. Solving for k, 6+1 = 2k-k, so k = 7.

AI explanation

Using Vieta's formulas for the equation x^2 - (k+6)x + 2(2k-1) = 0, the sum of the roots is k + 6 and the product of the roots is 4k - 2. The problem states that the sum is half of the product, which gives the equation k + 6 = 1/2 * (4k - 2). Multiplying both sides by 2 to clear the fraction results in 2k + 12 = 4k - 2. Solving for k, we subtract 2k from both sides and add 2 to both sides to get 14 = 2k, meaning k = 7.

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