quadratic equations

Question

Write the quadratic equation whose roots are 1 and -6 , and whose leading coefficient is 3 .

Answer 1

Ignoring for the moment the leading coefficient.....

If the roots are 1 and -6, then factors are (x - 1) and (x + 6). Multiply to get:

x^2 + 5x -6 = 0 (this is the "standard form" for a quadratic equation)

The "brute force" way to get the desired leading coefficient is to multiply everything by 3:

3x^2 + 15x - 18 = 0

Is it possible to do it another way?

Suppose one of the factors was 3x - 3

Now we have (3x - 3)(x + 6) <<This gives the same answer>>

Answer 2

Hi there! I'd love to help you out :)

For this one, we know x = 1 and x = -6. This means the factors of the polynomial are:

x - 1 and x + 6

Now, we also know we have a leading coefficient of 3. So when we multiply the factors we need to put a 3 in front:

3(x - 1)(x + 6)

Now just FOIL and simplify:

3 (x² +6x - x - 6)

3(x² + 5x - 6)

Now distribute the 3:

3x² + 15x - 18

There you have it! I hope this explanation was thorough. If you have any additional questions please don't hesitate to reach out. Learn on! :)