Palm Beach Community College s1

1.1 EQUATIONS Linear Equations is of the form ax + b = c, where a, b, and c are constants and a  0. Strategy for solving linear equations: [1] Clear parentheses, using the distributive property, and clear fractions or decimals by multiplying both side by LCD or a power of 10; [2] collect like terms on each side; [3] use addition (subtraction) property of equality to get on one side all terms containing the variable for which the equation is being solved and all remaining terms on the other side of the equal side; [4] use multiplication (division) property of equality to get the variable by itself on one side; [5] check your answer. 3 5 5 1. 8x + 5 = 29 2. 4a + 5 = 7a – 19 3. x   4 8 6

4. 0.03x + 0.004 = 3.514 5. 3(2y + 3) = 7y – 5 6. 5x – (6x + 2) = -3(x + 5)

2 1 2 7. 4x 8  4 x  8 8. 0.03(2x – 4) = 0.06x + 8 9. x7  0.4( x  17.5) 3 5 5

7x  28 23.5 31.5 15 10.   3 11. 5.7x   0.587 12. 14.83x  x  12 x4 x  4 42.85 4.7 5.73 1.2 QUADRATIC EQUATIONS Quadratic Equations: These are second-degree equations, meaning the largest exponent is two. Standard form is ax2 + bx + c = 0, where a, b, and c are coefficients of x. Factoring Method: Zero Factor Property (Zero Product Principle): If two factors are multiplied and the product is zero, then one of the factors must equal zero: ab = 0  a = 0 or b = 0 or both. This property is used to solve quadratic equations that can be factored with zero on one side. To solve equations by factoring: 1. Write the equation with zero on one side, say the right. 2. Factor the left side. 3. Set each factor equal to zero. 4. Solve the simpler equations. 5. Check the answer in the original equation. Square Root Method: If x2 = p and p ≥ 0, then x =  p . This method is good when the side of the equation containing the variable is a perfect square. Completing the Square: Any equation can be turned into a perfect square trinomial by completing the square: [1] If the coefficient of x2 ≠ 0, divide the equation by that coefficient; [2] rewrite the equation in the form 2 b 2 x + bx = -c; [3] add ( /2) to both sides of the equation: [4] write the trinomial as the square of a binomial and simplify the other side; [5] solve the equation by the Square Root method. Quadratic Formula: Any quadratic equation can be solved with this formula. Write the b  b2  4 ac equation in standard form, and substitute a, b, and c into the formula: x  . 2a Solve by factoring: 1. (a – 6)(a + 5) = 0 2. t2 – 6t - 27 = 0 3. p2 – 42 = p

4. 5x3 = 125x 5. 2x2 + 18x = 0 6. (5x + 3)(x – 4) = -26

Solve by square root method: 2 2 2 2 2  5 7. x = 121 8. w + 49 = 0 9. (a – 2) = 8 10. w    3  9

Find the perfect square trinomial whose first two terms are given and factor: 6 11. m2 + 14m 12. w2 – 5w 13. p2  p 5 Solve by completing the square: 14. x2 – 6x – 7 = 0 15. x2 – 10x – 3 = 0 16. 4x2 – 6x + 1 = 0

Solve by Quadratic Formula: 1 1 17. 2x2 + 9x + 10 = 0 18. x2  x 1  0 6 2

19. 4 + 20x = -25x2 20. -6x4 – 4x3 – 10x2 = 0

Try these: 21. Solve by factoring: -6 – 5p = -6p2

22. Solve by square root method: (x – 3)2 = 22

23. Solve by completing the square: 2x2 – 3x – 1 = 0

24. Solve by quadratic formula: 2x2 = -3 – 5x