Write the quadratic equation whose roots are 1 and 2 , and whose leading coefficient 5 is .

(Use the letter x to represent the variable.)

Why did the quadratic equation go to the circus? Because it needed roots and a leading coefficient to show off! Alright, let's get down to business. The quadratic equation you're looking for is:

5x^2 - 15x + 10 = 0

Now that's a real showstopper!

To find the quadratic equation with roots 1 and 2 and a leading coefficient of 5, we can use the fact that the roots of a quadratic equation can be used to determine its equation in the form of (x - r1)(x - r2) = 0, where r1 and r2 are the roots.

Given that the roots are 1 and 2, the equation becomes:
(x - 1)(x - 2) = 0

Expanding the equation, we get:
x^2 - 3x + 2 = 0

To include the leading coefficient of 5, we can multiply every term by 5:
5(x^2 - 3x + 2) = 0

Simplifying further, the quadratic equation with the desired roots and leading coefficient is:
5x^2 - 15x + 10 = 0

To write a quadratic equation with given roots, we need to use the fact that the sum of the roots is equal to the negation of the coefficient of the quadratic term (x^2), and the product of the roots is equal to the constant term.

Let's consider the roots 1 and 2. The sum of the roots is 1 + 2 = 3, and the product of the roots is 1 * 2 = 2.

Now, let's write the quadratic equation. Since the leading coefficient is 5, we can start with the equation:

5(x^2 + bx + c) = 0

Now, we need to find the values of b and c based on the sum and product of the roots.

The sum of the roots is 3, so we have the equation:

- b/5 = 3

Solving for b, we get:

b = -15

The product of the roots is 2, so we have the equation:

c/5 = 2

Solving for c, we get:

c = 10

Now we can substitute the values of b and c into the equation:

5(x^2 - 15x + 10) = 0

Simplifying, we have:

x^2 - 15x + 10 = 0

Therefore, the quadratic equation with roots 1 and 2, and a leading coefficient of 5, is:

5(x^2 - 15x + 10) = 0

If this is your current topic of study, you MUST know this:

5(x-1)(x-2) = 0

The 5 would only become important if you have the function
f(x) = 5(x-1)(x-2)

For the quadratic equation 5(x-1)(x-2) = 0 it would have no effect on
the roots.