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True if both $a and $b are TRUE True if either $a or $b is TRUE True if either $a or $b is TRUE, but not both True if $a is not TRUE True if both $a and $b are TRUE True if either $a or $b is TRUE

This equation is equivalent to the other one To show this, we can derive one squared binomial from the other: (4x 5)2 = (4x 5)(4x 5) = ( 1)2(4x 5)(4x 5) = ( 1)(4x 5)( 1)(4x 5) = ( 4x + 5)( 4x + 5) = ( 4x + 5)2 This duplicity occurs with all squared binomials It simply comes out of the fact that ( 1)2 = 1 6 To find the root of the quadratic, we can start with either of the binomial factor equations we found Let s use the first one: (4x 5)2 = 0 Taking the square root of both sides, we obtain 4x 5 = 0 We can add 5 to each side, getting 4x = 5 Dividing through by 4 gives us the root x = 5/4 The solution set is {5/4} Let s plug the root into the original quadratic to be sure that it works: 16x 2 40x + 25 = 0 16 (5/4)2 40 (5/4) + 25 = 0 16 25/16 50 + 25 = 0 25 50 + 25 = 0 25 + 25 = 0 0=0 7 We want to morph the following quadratic so we can get the left side into a product of a binomial with itself x 2 + 6x 7 = 0 We can add 16 to each side to get x 2 + 6x + 9 = 16 The left side can now be factored into a square of a binomial: (x + 3)2 = 16

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8 Let s take the square root of both sides of the binomial factor equation from solution to Prob 7, remembering to include both the negative and positive results: [(x + 3)2]1/2 = (161/2) This simplifies to x + 3 = 4 which can be stated as the following pair of first degree equations: x+3=4 or x + 3 = 4

Table 9-3

The first of these solves to x = 1, and the second solves to x = 7 The roots of the quadratic are therefore x = 1 or x = 7, so the solution set is {1, 7} Let s check these roots in the original quadratic When x = 1, we get x 2 + 6x 7 = 0 1 +6 1 7=0 1+6 7=0 7 7=0 0=0

9:

When x = 7, we get x 2 + 6x 7 = 0 ( 7) + 6 ( 7) 7 = 0 49 + ( 42) 7 = 0 49 42 7 = 0 7 7=0 0=0

9 To determine how many real roots a quadratic has, we can calculate the discriminant For a quadratic of the form ax 2 + bx + c = 0 the discriminant is b2 4ac The equation of interest is 2x 2 + 3x + 35 = 0 Here, a = 2, b = 3, and c = 35 Therefore b 2 4ac = 32 4 ( 2) 35 = 9 ( 280) = 9 + 280 = 289

The for loop, which repeatedly executes a statement, looks like this in PHP:

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