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To solve a quadratic equation by factoring: 1.? Start with the equation in the form Be sure it is set equal to zero!2.Factor the left hand side (assuming zero is on the right)3.Set each factor equal to zero4.? Solve to determine the roots (the values of x)Let's examine some possible situations: Factoring with GCF(greatest common factor)Factoring with DOTS(difference of two squares)Factoring TrinomialsFind the largest value that can be factored from each of the elements of the expression.Notice how the answers (the roots) can also be observed as the locations where the graph crosses the x-axis.Look carefully at this example to refresh this process:In a quadratic equation in descending order with a leading coefficient of one, look for the product of the roots to be the constant tern and the sum of the roots to be the coefficient of the middle term.Notice how the answers (the roots) can also be observed as the locations where the graph crosses the x-axis.?Or Isolate the Variable(Square Root Property)?Factoring Harder Trinomials?Tricky One!!Using Your Graphing CalculatorIf the leading coefficient is not equal to 1, you must think more carefully about how to set up your factors.?Be sure to get the equation set equal to zero before you factor.Graph? 2x2 - 5x + 2 The graphing calculator can be a very helpful tool in solving quadratic equations andchecking factors.How to use your TI-83+/84+ graphing calculator ?with quadratic equations.Click calculator.The discriminant is the name given to the expression that appears under the square root (radical) sign in the quadratic formula.Quadratic Formula:?Discriminant?The discriminant tells you about the "nature" of the roots of a quadratic equation given that a, b and c are rational numbers.? It quickly tells you the number of real roots, or in other words, the number of x-intercepts, associated with a quadratic equation.? There are three situations: Value of the discriminantExample showing nature of roots of ax2 + bx + c = 0Graph indicating x-interceptsy = ax2 + bx + cPOSITIVEThere are two real roots.(If the discriminant is a perfect square, the two roots are rational numbers.? If the discriminant is not a perfect square, the two roots are irrational numbers containing a radical.)There are two x-intercepts.ZEROThere is one real root.(The root is repeated.)There is one x-intercept.NEGATIVEThere are two complex roots.?There are no x-intercepts.When the roots of a quadratic equation are imaginary,they always occur in conjugate pairs.?A root of an equation is a solution of that equation.? If a quadratic equation with real-number coefficientshas a negative discriminant,then the two solutions to the equation are complex conjugates of each other. (Remember that a negative number under a radical sign yields a complex number.)The discriminant is the? b2- 4ac? part of the quadratic formula (the part under the radical sign).? If the discriminant is negative, when you solve your quadratic equation the number under the radical sign in the quadratic formula is negative --- forming complex roots.???????????????????????????? Quadratic equation:?? ?????????????????????????????? Quadratic formula:??? ?? ? ?Example 1: Find the solution set of the given equation over the set of complex numbers.a = 1,???? b = -10,???? c = 34Pick out the coefficient values representing a, b, and c, and substitute into the quadratic formula, as you would do in the solution to any normal quadratic equation. Remember, when there is no number visible in front of the variable, the number 1 is there.?HINT: ?When the directions say:?Express over the set of complex numbers,look for a negative value under the radical sign. ??????Example 2: ?????????????? Find the solution set of the given equation over the set of complex numbers.a = 3,??? b? = -4,?? c? = 10????? Example 3: Find the solution set of the given equation and express its roots in a+bi form. ?? * Be sure to set the quadratic equation equal to 0.* Arrange the terms of the equation from the?? highest exponent to the lowest exponent.?? Solving Quadratic Equations by Completing the SquareTopic Index | Algebra2/Trig Index | Regents Exam Prep Center?Solving quadratic equations by completing the square is overpowered by an "offspring" of this process, namely, the quadratic formula.? The quadratic formula was derived by completing the square on a quadratic equation.? Once? the quadratic formula was derived, it was no longer necessary to use the process of completing the square to solve "each" quadratic equation.? Even though completing the square is often overlooked in favor of the quadratic formula, it is still a valuable skill that will be needed in other mathematical situations.? Therefore, it is worthwhile to "get our feet wet" with these easier examples of applying the process of completing the square.The process of completing the square was explained in the section entitled Complete the Square.Examples:1.? Keep all terms containing x on one side.? Move the constant to the right.Get ready to create a perfect square on the left.? Balance the equation.Take half of the x-term coefficient and square it.? Add this value to both sides.Simplify and write the perfect square on the left.Take the square root of both sides.? Be sure to allow for both plus and minus.Solve for x.?2.? Keep all terms containing x on one side.? Move the constant to the right.Get ready to create a perfect square on the left.? Balance the equation.Take half of the x-term coefficient and square it.? Add this value to both sides.Simplify and write the perfect square on the left.Take the square root of both sides.? Be sure to allow for both plus and minus.Represent the negative radical as an imaginary number and solve for x.3.? Keep all terms containing x on one side.? This equation is all set up to start.Divide all terms by 5 to create a leading coefficient of one.Prepare to get a perfect square on the left.? Balance the equation.Take half of the x-term coefficient and square it.? Add this value to both sides.SimplifySimplifyWrite the perfect square on the left.Take the square root of both sides.? Be sure to allow for both plus and minus.Solve for x.Name__________________________________ Date ______________________ Period __________Quadratics PracticeDirections: For each problem solve by factoring (if possible, if not write prime), quadratic formula, and completing the square.1. FactoringQuadratic FormulaCompleting the Square2. FactoringQuadratic FormulaCompleting the Square3. FactoringQuadratic FormulaCompleting the Square4. FactoringQuadratic FormulaCompleting the Square5. ?FactoringQuadratic FormulaCompleting the Square6. FactoringQuadratic FormulaCompleting the Square ................
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