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.

Find the sum of the roots of 2x^2-5x+3=0

• The sum of roots is -\frac ba:
• =-\frac{-5}2
• =\frac52
• Answer: sum of roots = \frac52

Find the sum of the roots of x^2+4x+4=0

• The sum of roots is -\frac ba:
• =-\frac41
• =-4
• Answer: sum of roots = -4

The sum of roots of the polynomial 3x^2+kx-9=0 is 7. Find the value of k.

• a=3, b=k, c=-9
• The sum of roots is 7, which represents -\frac ba:
  • 7=-\frac k3
  • k=-21
• Answer: k=-21

The sum of roots of the polynomial 2x^2+2mx+3=0 is -3. Find the value of m.

• 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

Product of roots

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.

Find the product of the roots of 2x^2-5x+3=0

• The product of roots is \frac ca:
• =\frac32
• Answer: product of roots = \frac32

Find the product of the roots of x^2+4x+4=0

• The product of roots is \frac ca:
• =\frac44
• =1
• Answer: product of roots = 1

The product of roots of the polynomial kx^2+10x+4 is \frac23. Find the value of k.

• 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

Evaluating other expressions

The equation x^2+5x+4=0 has roots p and q. Evaluate p^2+q^2

• 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

Summary of finding roots and coefficients