Find k if the sum of the zeroes of x2−(k+6)x+2(2k+1) is half of their product.
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Step-by-Step Solution
Step 1: Identify coefficients of the quadratic polynomial
A general quadratic polynomial is given by ax2+bx+c. We compare the given polynomial x2−(k+6)x+2(2k+1) with this general form to identify the coefficients a, b, and c.
Step 2: State formulas for sum and product of zeroes
For a quadratic polynomial ax2+bx+c, the sum of its zeroes is given by −ab, and the product of its zeroes is given by ac. These are fundamental properties of quadratic equations.
Step 3: Substitute coefficients into sum and product formulas
Using the coefficients a=1, b=−(k+6), and c=2(2k+1), we substitute these into the formulas. The sum of zeroes becomes k+6, and the product of zeroes becomes 2(2k+1).
Step 4: Formulate the equation based on the given condition
The problem states that the sum of the zeroes is half of their product. We set up an equation using this condition and the expressions we found for the sum and product of zeroes.
Step 5: Solve the equation for k
Now we simplify and solve the equation for k. First, simplify the right side of the equation. Then, rearrange the terms to isolate k on one side.