Creating Quadratic Functions (Equations) from Irrational Roots
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Given Rational Roots: ( –3 +/– Ö4 ) Locate Roots, Add & Subtract Roots, Create Equation
-4 -3 -2 -1 0 +1 +2 +3 +4
Roots: X = –3 – Ö4 X = –3 + Ö4
Sum & Product: Sum = – 6 Product = + 5
Equation (Function): X2 – ( R1 + R2 ) X + ( R1R2) = X2 – 6X + 5
Changing all signs generates an equivalent but negative quadratic: Y or F(X) = – X2 + 6X – 5
To do this opposite (–) generation change all signs of (X) terms then to check factor equation.
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Given Rational Roots: ( +2 +/– Ö3 ) Locate Roots, Add & Subtract Roots, Create Equation
-4 -3 -2 -1 0 +1 +2 +3 +4
Roots: X = +2 – Ö3 X = +2 + Ö3
Sum & Product: Sum = + 4 Product = + 7
Equation (Function): X2 – ( R1 + R2 ) X + ( R1R2) = X2 + 4X + 7
Changing all signs generates an equivalent but negative polynomial: Y or F(X) = – X2 – 4X – 7
To do this opposite ( – ) generation change all signs of (x) terms then to check factor equation.
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Creating Quadratic Functions with Irrational ( Pairs of Numbers )Roots tends to be more difficult
but using the handy dandy Abstract Trinomial generator X2 – ( R1 + R2 ) X + ( R1R2) does help!
It is derived from (X – R1 ) ( X – R2 ) = ( X2
– R1X – R2 X + R1
R2 ) = X2 – (
R1 + R2 ) X + R1R2.
Since
Given Roots of: R1 & R2 thus
X = + R1 and X = + R2 and therefore
(X – R1 ) ( X – R2 ).
Tom Love