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.

For a quadratic polynomial ax² + bx + c, the sum of its zeroes is given by - (b)/(a), and the product of its zeroes is given by (c)/(a). These are fundamental properties of quadratic equations.

Find $k$ if the sum of the zeroes of $x^2 - (k + 6)x + 2(2k + 1)$ is half of their product.

Answer: The value of $k$ is $5$ .

Step-by-step solution

Step 1: Identify coefficients of the quadratic polynomial

A general quadratic polynomial is given by $ax^2 + bx + c$ . We compare the given polynomial $x^2 - (k + 6)x + 2(2k + 1)$ 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 $ax^2 + bx + c$ , the sum of its zeroes is given by $- \frac{b}{a}$ , and the product of its zeroes is given by $\frac{c}{a}$ . 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.

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Find kkk if the sum of the zeroes of x2−(k+6)x+2(2k+1)x^2 - (k + 6)x + 2(2k + 1)x2−(k+6)x+2(2k+1) is half of their product.