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Name____________________________________ Date_________ Block_________ FFPC…Solving Quadratic Equations In general, a quadratic expression looks like this: ax2 + bx + c, where a ≠ 0. There are 5 ways to solve for x in a quadratic equation: a) factor it and use the zero product property (zpp), b) symbolically get x by itself, c) use the quadratic formula, d) complete the square, e) use tables/graphs. We will focus on the first three ways to solve. Here we go. Factor/ZPP Step 1) Make sure the equation equals 0 (move numbers/variables around). 2) Factor the equation (remember to pull out any greatest common factors first). If the equation does not factor, then this method is worthless. Move on to another method. 3) Once in factored form, figure out what will make each factor equal to 0… these numbers are your solutions. Example: (x+3)(x-8) = 0 (this equation is already equal to 0 and it is already factored). To make each factor equal to 0, x must be -3 or 8. These two numbers are my solutions. Symbolically Step 1) This method only works if there is an x2 term and NO OTHER x. If there is another x, this method is worthless. Move on to another method. 2) Isolate the x2 term. Move numbers/variables away from the x2 term to get it by itself. 3) Square root both sides of the equation. When you take the square root of x2 it becomes the absolute value of x, which looks like this: |x|. Remember, when solving an absolute value problem, set up two cases, one positive and one negative. Example: x2 + 5 = 21 x2 = 16 (subtract 5 from both sides; x2 term is now isolated) |x| = 4 (took the square root of both sides) x = 4 or -4 (two answers when solving for absolute value) Quadratic Formula Step 1) This method ALWAYS works, but it has many details to keep track of. 2) Make sure the equation equals 0 (move numbers/variables around first if need be). 3) Plug in numbers into the formula below (remember, a quadratic equation looks like this: ax2 + bx + c = 0, where a ≠ 0): “a” is the coefficient of the x2 term, “b” is the coefficient of the x term, “c” is the constant term. The ± symbol simply means there are two cases, one is the plus (+) case, and one is the minus (-) case. Example: • 2 Use the Quadratic Formula to solve x – 4x – 8 = 0. Looking at the coefficients, I see that a = 1, b = –4, and c simplify: = –8. I'll plug them into the Formula, and Then the solution is The above solution has two answers: 2 + 2√3 and 2 – 2√3. You can leave the answer like it is or you can use the decimal answers of 5.464 and -1.464. Looks fun, huh??? Now, you try these methods on the following problems… 3) x 2 = 21 4) a 2 = 4 Solve using the Factor/ZPP method 2 m 2 + 7 = 88 5) x + 89) = 28 6) 2n 2 = 10) −144−5x = −500 a. x 2 − 36 = 0 b. x 2 − 16x + 64 = 0 c. x(2x − 6) = 0 2 −7n 2 = −448 7) −6m 2 11) = −414 ANSWERS: 6, -6 2 8) 7x 2 = 12) −21 −2k = −162 ANSWER: 8 ANSWERS: 0, 3 Solve using the Symbolic method IV. the song!). x 2 −Solve 5 = by 73 using the Quadratic Formula (sing 16n 2 = 49 a. b.10) 9) m 2 + 713) = 88 −5x 214) = −500 11) −7n 2 = −448 ANSWERS: 8.83, -8.83 12) −2k 2 = −162 ANSWERS: 10, -10 -1- Solve using the Quadratic Formula method 13) x − 5 = 73 14) 16n 2 = 49 2 -1- ANSWERS: -3, 2 Kuta Software - Infinite Algebra 1 Use the notes above to help you with the following… Solving Quadratic Equations with Square Name_________________________ Roots Solve each equation by taking square roots. symbolically. 1) k 2 = 76 2) k 2 = 16 3) x 2 = 21 4) a 2 = 4 5) x 2 + 8 = 28 6) 2n 2 = −144 7) −6m 2 = −414 8) 7x 2 = −21 9) m 2 + 7 = 88 10) −5x 2 = −500 11) −7n 2 = −448 12) −2k 2 = −162 13) x 2 − 5 = 73 14) 16n 2 = 49 -1- Date________________ Solving Quadratic Equations by Factoring Date________________ P Solve each equation by factoring. 1) (k + 1)(k − 5) = 0 2) (a + 1)(a + 2) = 0 3) (4k + 5)(k + 1) = 0 4) (2m + 3)(4m + 3) = 0 5) x 2 − 11x + 19 = −5 6) n 2 + 7n + 15 = 5 7) n 2 − 10n + 22 = −2 8) n 2 + 3n − 12 = 6 9) 6n 2 − 18n − 18 = 6 10) 7r 2 − 14r = −7 Using the Quadratic Formula Date________________ Period____ Solve each equation with the quadratic formula. 1) m 2 − 5m − 14 = 0 2) b 2 − 4b + 4 = 0 3) 2m 2 + 2m − 12 = 0 4) 2x 2 − 3x − 5 = 0 5) x 2 + 4x + 3 = 0 6) 2x 2 + 3x − 20 = 0 7) 4b 2 + 8b + 7 = 4 8) 2m 2 − 7m − 13 = −10 -1-