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

(Use the letter x to represent the variable.)

Question

Write the quadratic equation whose roots are -6  and -3, and whose leading coefficient is 1. 

(Use the letter x to represent the variable.)

whpalmer4 · Answer

Okay, if you have a polynomial $P(x)$ with roots $r_1,r_2,...r_n$, you can write it as $$P(x) = a(x-r_1)(x-r_2)...(x-r_n)$$where $a$ is a constant used to fit it to a given point, or give one of the coefficients a specified value, otherwise 1.

Here you want the leading coefficient to be 1, so the value of $a$ will turn out to be 1.  Remember that when you multiply a bunch of binomials together, the coefficient of the leading term is the product of the coefficients of the leading terms of the binomials.  

Once you've written it out in factored form, just expand it by multiplying it out.

whpalmer4 · Answer

Hint: don't be tripped up by subtracting negative numbers...

anonymous · Answer

x^2+9x+18=0, @whpalmer4 ?

whpalmer4 · Answer

Yes, that is the correct answer!

whpalmer4 · Answer

Now, if we didn't have that specification of the leading term's coefficient being 1, we could have all sorts of different answers.  All of these curves have the same roots, but a different value of $a$: