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QUADRATIC EQUATIONS <br />MATH10 <br />ALGEBRA<br />Quadratic Equations (Algebra and Trigonometry, Young 2nd Edition, page 113-135) <br />
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Week 3 Day 3<br />GENERAL OBJECTIVE<br />At the end of the chapter the students are expected to:<br /><ul><li>Solve quadratic equations using different methods,
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Solve equations leading to quadratic equation, and
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Solve real-world problems that involve quadratic equation.</li></li></ul><li>Week 3 Day 3<br />TODAY’S OBJECTIVE<br />At the end of the lesson the students are expected to:<br /><ul><li> To distinguish between pure quadratic equation and complete quadratic equation,
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To determine the number and type of solutions or roots of a quadratic equation based on the discriminant,
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To solve quadratic equations by factoring, square root method, completing the square and quadratic formula. </li></li></ul><li>Week 3 Day 3<br />DEFINITION<br />QUADRATIC EQUATION<br />A quadratic equation in x is an equation that can be written in the standard form <br />where a, b, and c are real numbers and a 0 .<br />a represents the numerical coefficient of x2 , <br />b represents the numerical coefficient of x, and <br />c represents the constant numerical term. <br />Example:<br />
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Week 3 Day 3<br /><ul><li>Pure Quadratic Equation</li></ul>If b=0, then the quadratic equation is termed a "pure" quadratic equation. <br /> Example: 3x2 +6=0 <br /><ul><li>Complete Quadratic Equation</li></ul>If the equation contains both an x and x2 term, then it is a "complete" quadratic equation. <br />The numerical coefficient c may or may not be zero in a complete quadratic equation. <br /> Example: x2 +5x+6=0 and 2x2 - 5x = 0 <br />
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Week 3 Day 3<br />DEFINITION<br />DISCRIMINANT OF A QUADRATIC EQUATION<br />The term inside the radical, b2 -4ac, is called the discriminant.<br />The discriminant gives important information about the corresponding solutions or roots of <br />where a, b, and c are real numbers and a 0 .<br />Positive<br />Two distinct real roots<br />One real root (a double or repeated root)<br />Zero<br />Negative<br />Two complex roots(complex conjugates)<br />
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Week 3 Day 3<br />EXAMPLE<br />Determine the nature of roots of the following <br />quadratic equation.<br />
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Week 3 Day 3<br />DEFINITION<br />COMPLEX NUMBER<br /> A complex number is an expression of the form<br /> where a and b are real numbers and <br /> a is the real part and b is the imaginary part .<br />EXAMPLE<br />3<br />4<br />0<br />6<br />-7<br />0<br />-7<br />
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Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS<br />There are four algebraic methods of solving quadratic equation <br />in one variable, namely:<br /><ul><li> solution by factoring
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solution by quadratic formula</li></li></ul><li>Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS BY FACTORING<br />The Factoring Method applies the Zero Product Property whichstates that if the product of two ormore factors equals zero, then at least one of the factors equals zero. <br /> Thus if B·C=0, then B=0 or C=0 or both.<br />STEPS:<br />Write the equation in standard form ax2 + bx + c = 0.<br />Factor the left side completely.<br />Apply the Zero Product Property to find the solution set.<br />
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Week 3 Day 3<br />EXAMPLE<br />Solve the following equations.<br />
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Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS BY SQUARE ROOT METHOD<br />The Square Root Property states that if an expression squared is equal to a constant , then the expression is equal to the positive or negative square root of the constant.<br />
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Week 3 Day 3<br />EXAMPLE<br />Solve the following equations.<br />
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Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE<br />STEPS:<br />
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Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE<br />
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Week 3 Day 3<br />EXAMPLE<br />Solve the following equations.<br />
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Week 3 Day 3<br />SOLVING QUADRATIC EQUATIONS BY QUADRATIC FORMULA<br />THE QUADRATIC FORMULA<br />The roots of the quadratic equation ax2 + bx + c = 0, where a, b, and c are constants and a 0 are given by:<br /> <br /> <br /> <br />
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DERIVATION OF QUADRATIC FORMULA BY COMPLETING THE SQUARE<br />Consider the most general quadratic equation: <br />Solve by completing the square: <br />Divide the equation by the leading coefficient a.<br />2. Subtract from both sides.<br />3. Subtract half of and add<br /> the result to both sides.<br />4. Write the left side of the equation as a perfect square and the right side as a single fraction.<br />
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DERIVATION OF QUADRATIC FORMULA BY COMPLETING THE SQUARE<br />Solve using the square root method.<br />6. Subtract from both sides<br /> and simplify the radical.<br />7. Write as a single fraction.<br />8. We have derived the quadratic formula.<br />
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Week 3 Day 3<br />EXAMPLE<br />Solve the following equations using the quadratic formula.<br />
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Week 3 Day 3<br />SUMMARY <br /><ul><li>The four methods for solving quadratic equations are: </li></ul> 3. completing the square<br />1. factoring <br />2. square root method<br />4. quadratic formula<br /><ul><li>Factoring and the square root method are the quickest and easiest but cannot always be used.
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Quadratic formula and completing the square work for all </li></ul> quadratic equations and can yield three types of solutions:<br /> two distinct real roots<br />one real root (repeated)<br />or two complex roots (conjugates of each other)<br />
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Week 4 Day 1<br />EQUATIONS IN QUADRATIC FORM<br />(OTHER TYPES) <br />
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TODAY’S OBJECTIVE<br />Week 4 Day 1<br />At the end of the lesson the students are expected to:<br /><ul><li>To find the sum and product of roots of a quadratic equation.
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To find the quadratic equation given the roots.
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To transform a difficult equation into a simpler linear or quadratic equation,
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To recognize the need to check solutions when the transformation process may produce extraneous solutions,
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To solve radical equations.</li></li></ul><li>Week 4 Day 1<br />RECALL <br /><ul><li>The four methods for solving quadratic equations are: </li></ul> 3. completing the square<br />1. factoring <br />2. square root method<br />4. quadratic formula<br /><ul><li>Factoring and the square root method are the quickest and easiest but cannot always be used.
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Quadratic formula and completing the square work for all </li></ul> quadratic equations and can yield three types of solutions:<br /> two distinct real roots<br />one real root (repeated)<br />or two complex roots (conjugates of each other)<br />
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SUM AND PRODUCT OF ROOTS<br /> Recall from the quadratic formula that when <br />Week 4 Day 1<br />
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Week 4 Day 1<br />SUM OF ROOTS<br />Sum of roots = r + s<br />
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Week 4 Day 1<br />PRODUCT OF ROOTS<br />Product of roots = (r) (s)<br />
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Week 4 Day 1<br />EXAMPLE<br />Determine the value of k that satisfies the given condition<br />
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Week 4 Day 1<br />FINDING THE QUADRATIC EQUATION GIVEN THE ROOTS <br />Example: Find the quadratic equations with the given roots.<br />
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Week 4 Day 1<br />RADICAL EQUATIONS<br />Radical Equations are equations in which the variable is<br />inside aradical (that is square root, cube root, or higher <br />root). <br />
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Week 4 Day 1<br />RADICAL EQUATIONS<br />Steps in solving radical equations:<br />Isolate the term with a radical on one side.<br />Raise both (entire)sides of the equation to the power that will eliminate this radical and simplify the equation.<br />If a radical remains, repeat steps 1 and 2.<br />Solve the resulting linear or quadratic equation.<br />Check the solutions and eliminate any extraneous solutions.<br />Note: When both sides of the equations are squared extraneous solutions can arise , thus checking is part of the solution. <br />
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EXAMPLE<br />Week 4 Day 1<br />Solve the following equations.<br />
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Steps in solving radical equations:</li></ul>Isolate the term with a radical on one side.<br />Raise both (entire)sides of the equation to the power that will eliminate this radical and simplify the equation.<br />If a radical remains, repeat steps 1 and 2.<br />Solve the resulting linear or quadratic equation.<br />Check the solutions and eliminate any extraneous solutions.<br />
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TODAY’S OBJECTIVE<br />Week 4 Day 2<br />At the end of the lesson the students are expected to:<br /><ul><li>To solve equations that are quadratic in form,
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To realize that not all polynomial equations are factorable.
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To solve equations that are factorable.</li></li></ul><li>Week 4 Day 2<br />EQUATIONS QUADRATIC IN FORM: u-SUBSTITUTION<br />Equations that are higher order or that have fractional powers often can be transformed into quadratic equation by introducing a u-substitution, thus the equation is in quadratic form.<br />Example:<br />
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Week 4 Day 2<br />EQUATIONS QUADRATIC IN FORM: u-SUBSTITUTION<br />Steps in solving equations quadratic in form:<br />Identify the substitution.<br />Transform the equation into a quadratic equation.<br />Apply the substitution to rewrite the solution in terms the original variable.<br />Solve the resulting equation.<br />Check the solution in the original equation.<br />
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EXAMPLE<br />Week 4 Day 2<br />Solve the following equations.<br />
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Week 4 Day 2<br />FACTORABLE EQUATIONS <br /><ul><li>EQUATIONS WITH RATIONAL EXPONENTS BY FACTORING
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POLYNOMIALEQUATION USING FACTORING BY GROUPING</li></li></ul><li>Week 4 Day 2<br />SUMMARY<br />Radical equations, equations quadratic in form, and factorable equations can often be solved by transforming them into simpler linear or quadratic equations. <br /><ul><li>Radical Equations: Isolate the term containing a radical and raise </li></ul> it to the appropriate power that will eliminate the radical. If there <br />is more than one radical, it does not matter which radical is isolated first. Raising radical equations to powers may cause extraneous solutions, so check each solutions.<br /><ul><li>Equations quadratic in form: Identify the u-substitution that transforms the equation into a quadratic equation. Solve the quadratic equation and then remember to transform back to the original equation.
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Factorable equations: Look for a factor common to all terms or factor by grouping. </li></li></ul><li>APPLICATION PROBLEMS<br />
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Week 4 Day 3<br />Start<br />RECALL<br />A<br />Read and analyze the problem<br />Make a diagram or sketch if possible<br />Solve the equation<br />Determine the unknown quantity. <br />Check the solution <br />Set up an equation, <br />assign variables to <br />represent what you<br /> are asked to find.<br />no<br />Is the unknown <br />solved?<br />no<br />yes<br />yes<br />Did you set up <br />the equation?<br />A<br />End<br />
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Week 4 Day 3<br />APPLICATION PROBLEMS<br />1. If a person drops a water balloon off the rooftop of an 81 foot building, the height of the water balloon is given by the equation where t is in seconds. When will the water balloon hit the ground?<br />(Classroom example 1.3.12 page 122)<br />You have a rectangular box in which you can place a 7 foot long fishing rod perfectly on the diagonal. If the length of the box is 6 feet, how wide is that box?<br />(Classroom example 1.3.13 page 123)<br />3. A base ball diamond is a square. The distance from base to base is 90 feet. What is the distance from the home plate to the second base? <br /> (#108 page 125)<br />
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Week 4 Day 3<br />4. Lindsay andKimmie, working together, can balance the financials for the Kappa Kappa Gama sorority in 6days. Lindsay by herself can complete the job in 5days less than Kimmie. How long will it take Lindsay to complete the job by herself? (# 113 page 125)<br />5.A rectangular piece of cardboard whose length is twice its width is used to construct an open box. Cutting a I foot by 1 foot square off of each corner and folding up the edges will yield an open box. If the desired volume is 12 cubic feet, what are the dimensions of the original piece of cardboard? (# 110 page 125)<br />6.Aspeed boat takes 1 hour longer to go 24 miles up a river than to return. If the boat cruises at 10mph in still water, what is the rate of the current? (#140 page 126)<br />
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Week 4 Day 3<br />7.Cost for health insurance with a private policy is given by<br /> where C is the cost per day and a is the insured’s age in years. Health insurance for a six year old, a=6, is $4 a day (or $1,460 per year). At what age would someone be paying $9 a day (or $3,285 per year).<br />(#73 page 134)<br />8. The period (T) of a pendulum is related to the length (L) of the pendulum and acceleration due to gravity (g) by the formula <br /> . If the gravity is and the period is 1 second find the<br /> approximate length of the pendulum. Round to the nearest inch. <br /> (#80 page 134)<br />
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