Question 30

Explanation

Given two quadratic functions

Finding the condition for intersection

Eliminate $y$ from both functions by equating $f(x) = g(x)$.

The condition for both functions to intersect at two different points is that the discriminant must be positive, namely $D > 0$.

From the inequality $(k + 5)(k + 1) > 0$, we get $k < -5$ or $k > -1$.

Since $k \neq 1$ and $k \neq 2$, the smallest positive integer that satisfies the condition is $k = 3$.

Substituting the value of k

Substitute the value $k = 3$ into the first equation to find the sum of roots $x$.

From this quadratic equation, the sum of its roots is

Finding the value of y

Substitute the value $k = 3$ into functions $f(x)$ and $g(x)$.

For $f(x)$

For $g(x)$

Then find the value of $x$ by eliminating $x^2$.

Substitute $x = 13 - y$ into $g(x)$.

From this quadratic equation, the sum of its roots is

Forming the new quadratic equation

The new quadratic equation with roots $x_1 + x_2$ and $y_1 + y_2$ is

Therefore, the quadratic equation with roots $x_1 + x_2$ and $y_1 + y_2$ is $x^2 - 26x - 56 = 0$.