In $x^2 - Bx + C = 0$, if the roots are $rt + su$ and $ru + st$, what is $C$ in terms of these roots? Write $C$ as a product involving $rt + su$ and $ru + st$.

After you write $C = (rt + su)(ru + st)$, expand it and see if you can factor your result to use $r + s$, $rs$, $t + u$, and $tu$ instead of $r,s,t,u$ individually.

When you get expressions like $r^2 + s^2$ or $t^2 + u^2$, remember that $a^2 + b^2 = (a + b)^2 - 2ab$. Use the sums and products you already found to compute these.

Use the formula $C = tu(r^2 + s^2) + rs(t^2 + u^2)$ along with $r + s = 10$, $rs = 1$, $t + u = 2$, and $tu = -3$, and the identities $r^2 + s^2 = (r + s)^2 - 2rs$ and $t^2 + u^2 = (t + u)^2 - 2tu$ to get a purely numerical expression for $C$.

In Desmos, type the single expression

$(-3)\cdot((10)^2 - 2\cdot1) + 1\cdot((2)^2 - 2\cdot(-3))$

as one line. Desmos will output a number; that value is $C$.

For any quadratic $x^2 - Sx + P = 0$ with roots $a$ and $b$, we have:

• $a + b = S$
• $ab = P$

Apply this to each given quadratic:

1. For $x^2 - 10x + 1 = 0$ with roots $r$ and $s$:

   • $r + s = 10$
   • $rs = 1$

2. For $x^2 - 2x - 3 = 0$ with roots $t$ and $u$:

   • $t + u = 2$
   • $tu = -3$

For the quadratic $x^2 - Bx + C = 0$ with roots $rt + su$ and $ru + st$, the constant term $C$ equals the product of the roots:

Start with

Expand this expression:

Group like terms:

• The terms with $tu$: $r^2tu + s^2tu = tu(r^2 + s^2)$.
• The terms with $rs$: $rst^2 + rsu^2 = rs(t^2 + u^2)$.

So we get:

Use the identity $a^2 + b^2 = (a + b)^2 - 2ab$.

1. For $r$ and $s$:

2. For $t$ and $u$:

Now substitute these into the expression for $C$ along with $tu = -3$ and $rs = 1$:

Evaluate the expression from the previous step:

So the value of $C$ is $-284$.