To find the factored form of a quadratic equation given the $x$-values (roots) $5\sqrt{2}$ and $-3 - \sqrt{3}$, we can follow these steps:

Step 1: Use the factored form of a quadratic equation

The factored form of a quadratic equation given two roots $r_1$ and $r_2$ is:

$f(x) = (x - r_1)(x - r_2)$

Here, $r_1 = 5\sqrt{2}$ and $r_2 = -3 - \sqrt{3}$.

Step 2: Substitute the roots into the equation

We substitute the given roots:

$f(x) = (x - 5\sqrt{2})(x - (-3 - \sqrt{3}))$

This simplifies to:

$f(x) = (x - 5\sqrt{2})(x + 3 + \sqrt{3})$

Step 3: Expand the factored form

Now, let's expand the expression:

$f(x) = (x - 5\sqrt{2})(x + 3 + \sqrt{3})$

To expand, distribute each term in the first factor over the second factor:

$f(x) = x(x + 3 + \sqrt{3}) - 5\sqrt{2}(x + 3 + \sqrt{3})$

Expand the terms:

$f(x) = x^2 + 3x + x\sqrt{3} - 5\sqrt{2}x - 15\sqrt{2} - 5\sqrt{6}$

Final Expression:

So, the expanded quadratic equation in standard form is:

$f(x) = x^2 + (3 + \sqrt{3} - 5\sqrt{2})x - 15\sqrt{2} - 5\sqrt{6}$

Would you like me to break down any of these steps further, or do you have any questions?

Related Questions:

1. What is the process to find the roots of a quadratic equation using the quadratic formula?
2. How can we simplify the expression further when dealing with square roots?
3. How does the discriminant help in finding the nature of the roots?
4. What is the geometric meaning of the roots of a quadratic equation?
5. How do you derive the general form of a quadratic equation from its factored form?

Tip:

Always double-check the signs of the roots when substituting them into the factored form, as it can affect the final equation!