Quadratic roots and coefficients

The roots in polynomials are closely linked to the coefficients of that polynomial.

All polynomials in this chapter will be written as follows - the letters are important here:

ax^2+bx+c

Sum of roots

If p and q are roots of ax^2+bx+c=0, then p+q=-\frac ba.

This means that the sum of roots of a quadratic is equal to -\frac ba.

a=2, b=2m, c=3
The sum of roots is -3, which represents -\frac ba:
- -3=-\frac{2m}2
- -3=-m
- m=3
Answer: m=3

If p and q are roots of ax^2+bx+c=0, then p\times q=\frac ca (positive).

In words, the product of roots of a quadratic is equal to \frac ca.

a=k, b=10, c=4
The product of roots is \frac23, which represents \frac ca:
- \frac23=\frac4k
- 2k=12
- k=6
Answer: k=6

The only things we know about p and q are:
- p+q=-\frac ba=-5
- pq=\frac ca=4
We need to rewrite p^2+q^2 in terms of p+q and pq only:
- p^2+q^2=(p+q)^2-2pq
Now we can substitute the values we know in!
- =(-5)^2-2(4)
- =25-8
- =17
Answer: p^2+q^2=17

	Coefficient link	Positive/Negative
Sum of roots	-\frac ba	Negative
Product of roots	\frac ca	Positive

Question	Answer
p+q	-\frac{b}{a}
p \times q	\frac{c}{a}
Sum of roots of 2x^2-5x+3=0	\frac{5}{2}
Sum of roots of x^2+4x+4=0	-4
If sum of roots of 3x^2+kx-9=0 is 7, find k	k = -21
If sum of roots of 2x^2+2mx+3=0 is -3, find m	m = 3
Product of roots of 2x^2-5x+3=0	\frac{3}{2}
Product of roots of x^2+4x+4=0	1
If product of roots of kx^2+10x+4=0 is \frac{2}{3}, find k	k = 6
Evaluate p^2+q^2 for x^2+5x+4=0 with roots p, q	17
How do you express p^2+q^2 in terms of sum and product of roots?	p^2+q^2 = (p+q)^2 - 2pq
What is the coefficient link for the sum of roots of a quadratic?	-\frac{b}{a}, negative
What is the coefficient link for the product of roots of a quadratic?	\frac{c}{a}, positive