99 Percentile SAT Math - Algebra, Sets & Venn Diagrams, Geometry & Coordinate Geometry by Chris B

Author:Chris B [B, Chris]
Language: eng
Format: mobi
Published: 2012-06-23T16:00:00+00:00

∴ The expression x2 – 5x – 14 0 will be negative or equal to zero when – 7 < x < 2.

For a quadratic expression of the form

ax2 + bx + c, (b2 - 4ac) is defined as the discriminant of the expression and is often denoted as D. i.e. D = b2 – 4ac

The following cases are possible for the value of the quadratic expression:

Case 1: If D < 0

i) If a < 0 then ax2 + bx + c < 0 for all x

ii) If a > 0 then ax2 + bx + c > 0 for all x.

In other words, we can say that if D is negative then the values of the quadratic expression takes the same sign as the coefficient of x2

This can also be said as;

If D < 0 then all real values of x are solutions of the inequalities ax2 + bx + c > 0 and ax2 + bx + c ≥ 0 for a > 0 and have no solution in case a < 0.

Also, for D < 0, all real values of x are solutions of the inequalities ax2 + bx + c < 0 and ax2 + bx + c ≤ 0 if a < 0 and these inequalities will not give any solution for a > 0.

Case 2: D = 0

If the discriminant of a quadratic expression is equal to zero, then the value of the quadratic expression takes the same sign as that of the coefficient of x (except when x = -b/2a at which point the value of the quadratic expression becomes 0).

We can also say the following for D = 0:

i) The inequality ax2 + bx + c > 0 has as a solution any x≠ - (b/2a) if a > 0 and has no solution if a < 0.

ii) The inequality ax2 + bx + c < 0 has as a solution any x ≠ - (b/2a) if a < 0 and has no solution if a > 0.

iii) The inequality ax2 + bx + c ≥ 0 has as a solution any x if a > 0 and has a unique solution x = -b/2a if a < 0.

iv) The inequality ax2 + bx + c ≤ 0 has as a solution any x if a < 0 and has a unique solution x = -b/2a for a > 0.

Case 3: D > 0

If x1 and x2 are the roots of the quadratic expression then it can be said that:

i) For a > 0, ax2 + bx + c is positive for all values of x outside the interval [x1, x2] and is negative for all values of x within the interval (x1, x2) Besides for values of x = x1 or x = x2 the value of the quadratic expression becomes zero (By definition of the root).

ii) For a < 0, ax2 + bx + c is negative for all values of x outside the interval [x1, x2] and is positive

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