Quadratic pattern:
A polynomial follows a quadratic pattern if it can be reduced to the form: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐.
This can be done by a simple substitution. For example the expression (𝑎𝑥 + 𝑏)2 can be
reduced to 𝑢2 by letting 𝑢 = 𝑎𝑥 + 𝑏. And 𝑥 4 can be reduced to 𝑢2 by letting 𝑢 = 𝑥 2 .
Example: Solve the following equations by factoring:
a) −2(𝑥 + 3)2 + 12(𝑥 + 3) + 14 = 0
b) 4(5𝑥 − 3)2 + 10(5𝑥 − 3) − 6 = 0
c) 16(3𝑥 + 7)2 − 40(3𝑥 + 7) + 25 = 0
1
, d) 9(2𝑥 + 1)2 − 4(𝑥 − 2)2 = 0
e) 36𝑥 4 − 12𝑥 2 + 1 = 0
f) 𝑥 6 + 7𝑥 3 − 18 = 0
g) 16𝑥 4 − 81 = 0
h) 5𝑥 8 + 8𝑥 4 − 4 = 0
2
A polynomial follows a quadratic pattern if it can be reduced to the form: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐.
This can be done by a simple substitution. For example the expression (𝑎𝑥 + 𝑏)2 can be
reduced to 𝑢2 by letting 𝑢 = 𝑎𝑥 + 𝑏. And 𝑥 4 can be reduced to 𝑢2 by letting 𝑢 = 𝑥 2 .
Example: Solve the following equations by factoring:
a) −2(𝑥 + 3)2 + 12(𝑥 + 3) + 14 = 0
b) 4(5𝑥 − 3)2 + 10(5𝑥 − 3) − 6 = 0
c) 16(3𝑥 + 7)2 − 40(3𝑥 + 7) + 25 = 0
1
, d) 9(2𝑥 + 1)2 − 4(𝑥 − 2)2 = 0
e) 36𝑥 4 − 12𝑥 2 + 1 = 0
f) 𝑥 6 + 7𝑥 3 − 18 = 0
g) 16𝑥 4 − 81 = 0
h) 5𝑥 8 + 8𝑥 4 − 4 = 0
2
