Section 1.5 Ã¢Â€Â“ Solve Quadratic Equations - McGraw-Hill Ryerson

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$Math and Chemistry - McGraw-Hill Ryerson$

Example 1Select a Strategy to Solve a Quadratic Equationa) Solve 2x 2 12x 14 5 0 byi) completing the squareii) using a graphing calculatoriii) factoringiv) using the quadratic formulab) Which strategy do you prefer? Justify your reasoning.Solutiona) i) 2x 2 12x 14 5 0x 2 6x 7 5 0x 2 6x 9 9 7 5 0(x 3) 2 16 5 0(x 3) 2 5 16x 3 5 4 or x 3 5 4The solutions are x 5 7 and x 5 1.ii) • Use the window settings shown.• Graph Y1 5 2x 2 12x 14.Divide both sides by 2.Take the square root of both sides.Technology TipRefer to the UseTechnology feature onpage 33 to see how tofind zeros using aTI-Nspire TM CASgraphing calculator.• Use the Zero operation to find the x-intercepts.The solutions are x 5 1 and x 5 7.iii) 2x 2 12x 14 5 0x 2 6x 7 5 0(x 7)(x 1) 5 0x 7 5 0 or x 1 5 0x 5 7 or x 5 1Divide both sides by 2.Find the binomial factors of the trinomialx 2 — 6x — 7.44 MHR • Functions 11 • Chapter 1
iv) 2x 2 12x 14 5 0x 2 6x 7 5 0a 5 1, b 5 6, ________ and c 5 7.x 5 b ____ b 2 4ac 2a________________5 (6) ______ (6) 2 4(1)(7) ___ 2(1)5 __6 64 25 __ 6 82 5 14_ or 2_2 2 5 7 or 1Divide both sides by 2.Substitute the values of a, b, and c into thequadratic formula and simplify.b) While all four methods produce the same solutions, factoring isprobably the best strategy for this example. The quadratic expressionis easy to factor, so this method is the fastest. If the quadraticexpression could not be factored, either the graphing calculatormethod or using the quadratic formula would be preferred.ConnectionsIn this example, theroots are integers.However, manyquadratic equationshave irrational roots.If exact roots areasked for, then eithercompleting the squareor the quadratic formulais a better methodto use. The graphingcalculator methodwill only provideapproximations.Solving 2x 2 12x 14 5 0 is equivalent to finding the zeros, orx-intercepts, of the function f (x) 5 2x 2 12x 14. The two solutionsin Example 1 represent the two x-intercepts of the functionf (x) 5 2x 2 12x 14. However, not all quadratic functions havetwo x-intercepts. Some have one x-intercept, while others have nox-intercepts. The next example illustrates this.Example 2Connect the Number of Zeros to a GraphFor each quadratic equation given in the form ax 2 bx c 5 0, graphthe related function f (x) 5 ax 2 bx c using a graphing calculator.State the number of solutions of the original equation. Justify eachanswer.a) 2x 2 8x 5 5 0b) 8x 2 11x 5 5 0c) 4x 2 12x 9 5 0Solutiona) The parabola opens downward and the vertexis located above the x-axis, so the function hastwo zeros.The equation 2x 2 8x 5 5 0 has twosolutions.1.5 Solve Quadratic Equations • MHR 45
Page 4 and 5: ) The parabola opens upward and the
Page 6 and 7: Since the discriminant is greater t
Page 8 and 9: 8. Determine the value(s) of k for

Section 1.5 Ã¢Â€Â“ Solve Quadratic Equations - McGraw-Hill Ryerson

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