Roots of quadratic equations

1
Roots of quadratic equations

• Investigating the relationships between the roots
  and the coefficients of quadratic equations.

2
Investigating roots

• Solve each of the following quadratic equations
• a) x2 7x 12 0
• b) x2 5x 6 0
• c) x2 x 20 0
• d) 2x2 5x 3 0
• Write down the sum of the roots and the product
  of the roots.
• Roots of polynomial equations are usually denoted
  by Greek letters.
• For a quadratic equation we use alpha (a) beta
  (ß)

Investigate roots (excel)
3
Properties of the roots of polynomial equations

• ax2 bx c 0
• a(x - a)(x - ß) 0 a 0
• This gives the identity
• ax2 bx c a(x - a)(x - ß)
• Multiplying out
• ax2 bx c a(x2 ax ßx aß)
• ax2 aax aßx aaß
• ax2 ax(a ß) aaß
• Equating coefficients
• b a(a ß) c aaß
• -b/a a ß c/a aß

4
Task

• Use the quadratic formula to prove the results
  from the previous slide.
• -b/a a ß c/a aß

5
Properties of the roots of polynomial equations

• Find a quadratic equation with roots 2 -5
• -b/a a ß c/a aß
• -b/a 2 -5 c/a -5 2
• -b/a -3 c/a -10
• Taking a 1 gives b 3 c -10
• So x2 3x 10 0
• Note There are infinitely many solutions to this
  problem.
• Taking a 2 would lead to the equation 2x2 6x
  20 0
• Taking a 1 gives us the easiest solution.
• If b and c are fractions you might like to pick
  an appropriate value for a.

6
Properties of the roots of polynomial equations

• The roots of the equation 3x2 10x 8 0 are a
  ß
• 1 Find the values of a ß and aß.
• a ß -b/a 10/3
• aß c/a -8/3
• 2 Find the quadratic equation with roots
  3a and 3ß.
• The sum of the new roots is 3a 3ß 3(a ß)
  3 10/3 10
• The product of the new roots is 9aß -24
• From this we get that 10 -b/a -24 c/a
• Taking a 1 gives b -10 c -24
• So the equation is x2 10x 24 0

7
Properties of the roots of polynomial equations

• 3 Find the quadratic equation with roots a 2
  and ß 2
• The sum of the new roots is a ß 4 10/3
  4 22/3
• The product of the new roots is
  (a 2)(ß 2) aß 2a 2ß 4
  aß 2(a ß) 4 -8/3 2(10/3) 4
  8
• So 22/3 -b/a 8 c/a
• To get rid of the fraction let a 3, so b -22
  c 24
• The equation is 3x2 22x 24 0

8
Properties of the roots of polynomial equations

• The roots of the equation x2 7x 15 0 are a
  and ß.
• Find the quadratic equation with roots a2 and ß2
• a ß 7 aß 15
• (a ß)2 49 a2ß2 225
• a2 2aß ß2 49
• a2 30 ß2 49
• a2 ß2 19
• So the equation is x2 19x 225 0

9
Try these

• Page 10 Exercise 1C