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| Algebraic Equations and Inequalities |
 Properties of Equations and Inequalities Graphical Interpretation of Solutions
Linear Equations Linear Inequalities
Linear Equations Involving Absolute Value Linear Equations Involving a Few Absolute Values Linear Inequalities Involving Absolute Value
 Quadratic
EquationsQuadratic Equations and Quadratic Functions Extreme Value of Quadratic Function Quadratic Formula Solving Quadratic Equations by Factoring Quadratic
Inequalities |
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let us consider a monic quadratic equation, x2
+ b x + c = 0.
If x1 and x2 are the roots of the equation, then the quadratic polynomial can be factored into two linear factors, x2
+ b x + c = (x
– x1)(x
– x2).
Expand the expression on the right side: x2
+ b x + c = x2
– x (x1
+ x2)
+ x1 x2.
Regroup the terms: (b
+ x1 + x2)
x + c = x1
x2.
This identity has to be true for any values of x.
c
= x1 x2. (*)
Therefore, (b
+ x1 + x2)
x = 0
for any values of x.
Hence,
x1 + x2
= – b. (**)
Formulas (*) and (**) can be used to find the roots of a quadratic equation as well as to check up whether some supposed roots are correct.
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