Page 113 - Algebra

Sum and product of roots
You can find the sum and product of the roots of a quadratic equation without solving the equation.
REMEMBER:
For a quadratic equation ax2 + bx + c = 0
−𝑏 Sumofroots= 𝑎
Product of roots = 𝑐 𝑎
Worked Example
Find the difference between the sum and the product of roots of the following quadratic equation: 2x2 + 8x = 9
Solution:
The equation can be written as:
2x2 +8x–9=0
Here a = 2, b = 8 , and c =– 9
−𝑏 −8 Sumofroots=𝑎=2 =
𝑐 −9 Product of roots = 𝑎 = 2
Difference =
=1 2
–4
−9 9 –4–(2)=–4+2
6.3. Identities
An identity is an equality that is true for every variable. For example,
(x + 1) (x – 2) = x (x – 2) + 1(x – 2)
= x2 – 2x + x – 2 = x2 – x – 2
Therefore, (x + 1) (x – 2) = x2 – x – 2 is an identity.
But,
x2 – x – 2 = 4, for x = 3
This is not an identity because it’s not true for every value of x.
REMEMBER:
(a + b)2 = a2 + 2ab + b2
(a–b)2 =a2 –2ab+b2
a2 – b2 = (a + b) (a – b)
(x + a) (x + b) = x2 + (a + b)x + ab
211212
x + 2 = (x + ) – 2 = (x – ) + 2
𝑥𝑥𝑥
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a – b)3 = a3 – 3a2b + 3ab2 – b3
a3 + b3 = (a + b)3 – 3ab(a + b) or (a + b) (a2 – ab + b2) a3 –b3 =(a–b)3 –3ab(a–b)or(a–b)(a2 +ab+b2)
x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx)
Page 112 of 177
Algebra I & II