Module 6: Notes and Solutions to Select Practice Problems. These solutions were prepared by Mr. John Hemmerling with revisions by Mr. Louis Aquila Practice Exercises-Unit 1. (3) -a - 3 = -10 Perform the opposite/inverse operation until you get the variable alone on one side of the equal sign. The opposite of -3 is +3 so: -a - 3 +3 = -10 +3 -a = -7 The variable is still not isolated. There is a minus sign next to it, so we are not finished. You can remove the minus sign by multiplying both sides of the equation by -1. (-1)(-a) =( -7)(-1) Thus, a = 7 Check: -a - 3 = -10 so -7 - 3 = -10 must be true.√ (14) 3e 9 5 There are two ways to solve this equation. 1) There are two operations here. The most 2) There is a fraction in front of the obvious is probably the division by 5. The variable. Multiply a fraction by its opposite of division is multiplication so reciprocal (Check out Module 5 in the multiply everything by 5. Course Guide if you want to review this topic) and you get 1. Multiplying by 1 gives you the variable by itself. Since this 5 3e 95 is an equation, you must multiply 5 everything by the reciprocal. -3e = -45 5 3e 5 9 The only operation left is multiplication by 3 5 3 (-3). The opposite of multiplication is division so divide everything by (-3) As you can see, the variable is alone now and all that's left is the simplification of the right-hand side (rhs) of the equation. 3e 45 3 3 9 5 45 e 15 Thus e = 15 3 3 Check: 3e 3(15) 45 9 so 9 5 5 5 (39) -14 = 5p + 18 This problem could just as easily been written as 5p + 18 = -14. Anyway, the solution of the original problem follows. -14 - 18 = 5p + 18 -18 - 32 5p so p = -32/5 5 5 32 14 5 18 Check: -14 = 5p + 18 so must be true. √ 5 14 32 18 (47) 7 3x 2 4 Observe that it makes no difference if the variable term is on the right or the left. 3x 2 4 3x 72 22 4 3x 9 4 4 3 x 4 9 3 4 3 12 xor x 12 7 Check: 7 3x 312 2 so 7 2 must be true. 4 4 36 7 2 √ 4 7 92 (60) 2b - 9 = 5b This problem has the variable b on both sides of the equal sign. You have to get them together to solve the equation. Adding the opposite of either term to both sides of the equation will do it. 2b - 2b - 9 = 5b - 2b -9 = 3b 9 3b so b = -3 3 3 Check: 2b - 9 = 5b so 2(-3) - 9 = 5(-3) must be true. -6 -9 = -15 √ (68) y - (2 + 2y) = 6 First remove the grouping symbols. Remember to distribute the negative sign over every term within the parentheses. Then you need to combine like terms. y - (2 + 2y) = 6 y - 2 - 2y = 6 -2 - y = 6 -2 + 2 - y = 6 + 2 -y = 8 so y = -8 Check: y - (2 + 2y) = 6 so -8 - (2 + 2[-8]) = 6 must be true. -8 - (2 + [-16]) = 6 -8 - (-14) = 6 -8 + 14 = 6 √ (74) 3( x 4) 9 2 First, multiply everything by 2 to remove the denominator. 3( x 4) 2 (2) 9 2 We now have the equation: -3(x - 4) = -18 -3x + 12 = -18 -3x + 12 - 12 = -18 - 12 -3x = -30 x = 10 Check: 3( x 4) 3(10 4) 9 so 9 must be true. 2 2 3(10 4) 9 2 30 12 9 √ 2 18 9 2 Notes for Unit 2. For the quadratic equation ax2 + bx + c = 0 (in standard form) and a 0 , the Quadratic Formula is. b b 2 4ac 2a To use this formula, put the numbers in for a, b and c and follow the order of operations. The Quadratic Formula can be used to find the solutions to all quadratic equations (in standard form). Each problem below will use the quadratic formula and where applicable we will also solve the equation by factoring. You do not have to solve a quadratic equation both ways. First try to factor the expression ax2 + bx + c. If you can do this, then you will be able to find the solutions without using the quadratic formula. If you cannot factor the expression ax2 + bx + c, then you have no recourse but to use the quadratic formula. Only use the quadratic formula if you cannot factor. Below where the expression ax2 + bx + c can be factored, we still solve the problem using the formula simply to convince you that both approaches give the same answers. x 4. x2 + 5x + 6 = 0 This quadratic equation is factorable. Use the Quadratic Formula: x2 + 5x + 6 = 0 a = 1, b = 5, c = 6 (x + 2)(x + 3) = 0 x In order for this product to be equal to zero: 5 5 2 4(1)(6) 5 25 24 2(1) 2 5 1 5 1 2 2 5 1 4 x 2 2 2 5 1 6 x 3 2 2 Check: x2 + 5x + 6 = 0 so (-2)2 + 5(-2) + 6 = 0 and (-3)2 + 5(-3) + 6 = 0. x (x + 2) = 0, so x = -2 (x + 3) = 0, so x = -3 4 + (-10) + 6 = 0 √ 9 + (-15) + 6 = 0 √ 11. 16t2 - 25 = 0 This quadratic equation is factorable. 16t2 - 25 = 0 (4t - 5)(4t + 5) = 0 In order for this product to be equal to zero: Use the Quadratic Formula: a = 16, b = 0, c = -25 0 0 2 4(16)( 25) 1600 x 2(16) 32 40 5 x 32 4 40 5 x 32 4 (4t - 5) = 0 4t - 5 + 5 = 0 + 5 4t = 5 so t = 5/4 OR (4t + 5) = 0 4t + 5 - 5 = 0 - 5 4t = -5 so t = -5/4 Check: 16t2 - 25 = 0 so 16(5/4)2 - 25 = 0 (Check is the same for x = -5/4 ) 16(25/16) - 25 = 0 25 - 25 = 0 √ 18. 2x2 - x = 0 This quadratic equation is factorable. Use the Quadratic Formula: 2x2 - x = 0 so x(2x - 1) = 0 a = 2, b = -1 , c = 0 In order for this product to be equal to zero: (1) (1) 2 4(2)(0) 1 1 2 ( 2 ) 4 x=0 11 OR x 0 2x - 1 = 0 4 2x - 1 + 1 = 0 + 1 11 1 x 2x = 1 so x = 1/2 4 2 2 2 2 Check: 2x - x = 0 so 2(0) - 0 = 0 √ and 2(1/2) - (1/2) = 0 x 1 1 2 0 4 2 √ 1 1 0 2 2 26. 8x2 + 3x - 12 = 0 The expression 8x2 + 3x - 12 cannot be factored (try it), so you must use the Quadratic Formula: a = 8, b = 3, c = -12 x 3 32 4(8)( 12) 3 9 384 3 393 2(8) 16 16 Thus, x 3 393 3 393 OR 16 16 Check: I'm only going to do one of these since the math is the essentially the same for both. 8x2 + 3x - 12 = 0 so 2 3 393 3 393 3 12 0 8 16 16 3 393 3 393 3 393 3 12 0 2 16 16 9 3 393 3 393 393 9 3 393 12 32 032 32 16 402 6 393 2 9 3 393 12(32) 0 402 6 393 (18) 6 393 384 0 402 402 0 Once again, if you can factor, then solve the quadratic equation by factoring. Otherwise, you must use the quadratic formula.

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