Quadratic equations and functions — Topic practice

−7

In f(x) = 3(x − 2)² − 7, the squared quantity is at least zero and its coefficient is positive. The smallest output occurs when x = 2, making the square zero. Thus the minimum value is −7, while the minimizing input is 2. Review 3.06 and 3.07.

9

The leading coefficient is 1, so this is genuinely quadratic for every c. One distinct real root requires

D = 6² − 4(1)c = 0, c = 9.

Equivalently, x² + 6x + 9 = (x + 3)², which vanishes only at −3. Review 3.11.

$\frac{\text{37}}{\text{4}}$

For 2x² − 7x + 3 = 0, the root sum is r + s = 7/2 and the root product is rs = 3/2. Hence

r² + s² = (r + s)² − 2rs = $\frac{\text{49}}{\text{4}}$ − 3 = $\frac{\text{37}}{\text{4}}$.

Check: the roots are 3 and 1/2, so their squared sum is 9 + 1/4. The symmetric-expression route avoids unnecessary root solving. Review 3.13.