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8a: Solving Quadratic Equations Using Square Roots Algebra 1 CCSS A.REI.4.b Essential Question: How do you solve a quadratic equation using square roots? Vocabulary ο· Quadratic Equation An equation whose highest degree is 2 or highest exponent is 2; usually written in the form of ππ₯ 2 + ππ₯ + π ο· Solution/ Roots/Zeros The solution of a quadratic equation where its parabola intersects the x-axis Skills Check β Simplify each expression. 1) β36 What are possible answers when simplifying a square root? 2) ββ81 49 3) β100 16 4) ββ121 You can solve equations with squared variables (i.e. π₯ 2 ) by using its inverse operation, square root βπ₯ 2 = ±π₯. Notice that the answer to a square root is the negative and positive of a number. β16 = ±4 β64 = ±8 β100 = ±10 Why is the answer to a square root the positive and negative of a number? ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ How do I simplify a radical expression when it is not a perfect square? A radical expression can be written as two separate radical expressions that are multiplied with each other. βππ = βπ β βπ To simplify a radical expression, remove perfect square factors of expression. β54 = β9 β β6 = ±3 β β6 = ±3β6 Practice β Simplify each radical expression. 1) β50 How do I solve quadratic equations using square roots? 2) β48 3) β5β300 4) β3β18 Solving a quadratic equation using square roots: 1. Isolate the squared variable using inverse operations. 2. Square root both sides of the equation. 3. Remember to include the positive and negative of your final answer. Practice β Solve each quadratic equation using square roots. a) π‘ 2 β 25 = 0 b) 2π2 + 32 = 0 c) 2π₯ 2 β 98 = 0 d) 3π2 + 12 = 12 When would a quadratic equation have no solution? Try this: π2 + 25 = 0 Explain why you could not determine the solution. ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ Practice β Solve each equation by finding the square roots. If the equation has no solution, write no solution. e) π2 β 225 = 0 f) π 2 + 25 = 25 g) π₯ 2 β 9 = β16 h) 27 β π¦ 2 = 0 i) Michaelβs work is shown below. Explain the error he made. ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ 8b: Solving Quadratic Equations by Factoring Algebra 1 CCSS A.REI.4.b Essential Question: How do I solve quadratic equations by factoring? Vocabulary: ο· Zero Product For every real number π and π, if ππ = 0, then π = 0 or π = 0. Property In other words, when multiplying two numbers, in order to get a product of 0, one of the two numbers have to be 0. Skills Check β Factor each quadratic expression. a) π₯ 2 + 5π₯ + 4 How do I solve equations using the Zero Product Property? b) 8π₯ 2 + 8π₯ β 6 1. Separate and set each binomial parenthesis equal to 0. 2. Solve each equation. 3. The answers represent the roots to the quadratic equation. Practice β Determine the roots for each equation. c) (π₯ + 5)(2π₯ β 6) = 0 d) β3π(2π β 10) = 0 How do I solve quadratic equations by factoring? You can also use the Zero Product Property to solve equations in the form of ππ₯ 2 + ππ₯ + π = 0 if the equation can be factored. 1. 2. 3. 4. 5. Make sure the equation is in standard form (ππ₯ 2 + ππ₯ + π = 0) Factor the quadratic equation. Use the Zero Product Property to separate and set each factor equal to 0. Solve each equation. The answers represent the roots of the quadratic equation. Practice β Determine the solution for each quadratic equation. e) π₯ 2 + 6π₯ + 8 = 0 f) 2π₯ 2 β 5π₯ = 88 g) 2π₯ 2 β 9π₯ = 0 h) 4π¦ 2 = 25 i) The sides of a small square are increased by 6 centimeters. The area of the new square is 81 square centimeters. Find the length of a side of the original square. j) You are building a rectangular pool. You want the area to be 120 ft. You also want the length of the pool to be 8 feet more than twice its width. Find the dimensions of the pool. Summary: In your own words, describe the Zero Product Property. ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ ____________________________________________________________ 8c: Completing the Square Algebra 1 CCSS A.REI.4.b Essential Question: Why do we complete the square to solve quadratic equations? Skills Check β Determine the solutions to the quadratic equation. a) π₯ 2 + 10π₯ + 25 = 121 How can you create a perfect square trinomial? You can make a quadratic equation a perfect square trinomial by completing the square. To make a perfect square trinomial, take half the value of b and square it: π 2 ( ) 2 Practice β Find the value that would complete the square. b) π₯ 2 β 6π₯ c) π¦ 2 + 8π¦ d) π₯ 2 + 9π₯ How do I find the roots of a quadratic equation by completing the square? 1. Isolate the π term if necessary. π 2 2. Find the value that would complete the square: (2) 3. Factor the perfect square trinomial. 4. Solve using square roots. Practice β Determine the solutions by completing the square. e) π 2 + 8π = 48 f) π 2 + 22π = β85 g) π₯ 2 β 6π₯ β 7 = 0 h) π₯ 2 + 8π₯ + 12 = 0 i) π₯ 2 + 10π₯ = β17 j) π₯ 2 β 2π₯ β 5 = 0 8d: The Quadratic Formula Algebra 1 CCSS A.REI.4.b Essential Question: How can you find the solution to a quadratic equation that canβt be factored? Vocabulary: ο· Quadratic Formula The quadratic formula is another method of finding the solution to a quadratic equation: Quadratic Formula: How do I use the quadratic formula? 1. Label the π, π, and π. 2. Substitute each value into the quadratic formula. 3. Simplify. Practice β Use the Quadratic Formula to determine the solution to each quadratic function. 1) π₯ 2 β 2π₯ β 8 = 0 2) π₯ 2 + 6 = 5π₯ 3) 2π₯ 2 + 4π₯ β 7 = 0 5) 7π₯ 2 β 2π₯ β 8 = 0 4) β3π₯ 2 + 5π₯ β 2 = 0 6) Which equation does 9±β(β9)2 β4(5)(β7) 2(5) to? give the solutions 8e: The Discriminant Algebra 1 CCSS A.REI.4.b Essential Question: What is the discriminant and what information does it provide? Vocabulary: ο· Discriminant The expression under the radical in the quadratic formula; the discriminant tells you the number of solutions to a quadratic equation A quadratic equation can have one solution, two solutions, or no solutions. How can I determine the number of solution/s 1) Substitute values of π. π, and π into the discriminant: π 2 β 4ππ a quadratic equation will have? ο· If π 2 β 4ππ > 0, then there are TWO solutions ο· If π 2 β 4ππ < 0, then there are NO solutions ο· If π 2 β 4ππ = 0, then there is ONE solution Practice β Determine how many times each quadratic equation intersects the xaxis. 1) 0 = 4π₯ 2 β 12π₯ = 9 2) 3π₯ 2 β 5π₯ = 1 3) 3π₯ 2 β 4π₯ = 7 4) π₯ 2 + π₯ = 0 5) Ken claims that the discriminant of 2π₯ 2 + 5π₯ β 1 = 0 is 17. What error did he make? ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ What is the correct discriminant?