ALevel数学2.ppt

Boardworks Ltd 20051 of 50 Boardworks Ltd 20051 of 50AS-Level Maths:Core 1for EdexcelC1.2 Algebra and functions 2This icon indicates the slide contains activities created in Flash.These activities are not editable.For more detailed instructions,see the Getting Started presentation.Boardworks Ltd 20052 of 50Contents Boardworks Ltd 20052 of 50Quadratic expressionsQuadratic expressionsFactorizing quadraticsCompleting the squareSolving quadratic equationsThe discriminantGraphs of quadratic functionsExamination-style questions Boardworks Ltd 20053 of 50Quadratic expressionsA quadratic expression is an expression in which the highest power of the variable is 2.For example:x2 2w2+3w+14 5g2 t22x is a variable.a is the coefficient of x2.b is the coefficient of x.c is a constant term.ax2+bx+c (where a 0)The general form of a quadratic expression in x is:Boardworks Ltd 20054 of 50Contents Boardworks Ltd 20054 of 50Factorizing quadraticsQuadratic expressionsFactorizing quadraticsCompleting the squareSolving quadratic equationsThe discriminantGraphs of quadratic functionsExamination-style questions Boardworks Ltd 20055 of 50Factorizing quadratic expressionsFactorizing an expression is the inverse of expanding it.Expanding or multiplying outFactorizingWhen we expand an expression we multiply out the brackets.(x+1)(x+2)x2+3x+2When we factorize an expression we write it with brackets.Boardworks Ltd 20056 of 50Factorizing quadratic expressionsQuadratic expressions of the form ax2+bx can always be factorized by taking out the common factor x.For example:No constant term3x2 5x=x(3x 5)When a quadratic has no term in x and the other two terms can be written as the difference between two squares,we can use the identityThe difference between two squaresa2 b2=(a+b)(a b)to factorize it.For example:9x2 49=(3x+7)(3x 7)Boardworks Ltd 20057 of 50Factorizing quadratic expressionsQuadratic expressions of the form x2+bx+c can be factorized if they can be written using brackets as(x+d)(x+e)where d and e are integers.If we expand(x+d)(x+e),we have(x+d)(x+e)=x2+dx+ex+de=x2+(d+e)x+de Quadratic expressions with a=1 Boardworks Ltd 20058 of 50Factorizing quadratic expressionsQuadratic expressions of the general form ax2+bx+c can be factorized if they can be written using brackets as(dx+e)(fx+g)where d,e,f and g are integers.If we expand(dx+e)(fx+g),we have(dx+e)(fx+g)=dfx2+dgx+efx+eg=dfx2+(dg+ef)x+eg The general form Boardworks Ltd 20059 of 50Contents Boardworks Ltd 20059 of 50Completing the squareQuadratic expressionsFactorizing quadraticsCompleting the squareSolving quadratic equationsThe discriminantGraphs of quadratic functionsExamination-style questions Boardworks Ltd 200510 of 50Perfect squaresSome quadratic expressions can be written as perfect squares.For example:x2+2x+1 =(x+1)2x2+4x+4 =(x+2)2x2+6x+9 =(x+3)2x2 2x+1 =(x 1)2x2 4x+4 =(x 2)2x2 6x+9 =(x 3)2How could the quadratic expression x2+8x be made into a perfect square?We could add 16 to it.In general:x2+2ax+a2 =(x+a)2 or x2 2ax+a2 =(x a)2 Boardworks Ltd 200511 of 50Completing the squareAdding 16 to the expression x2+8x to make it into a perfect square is called completing the square.x2+8x =x2+8x+16 16We can writeIf we add 16 we then have to subtract 16 so that both sides are still equal.By writing x2+8x+16 we have completed the square and so we can write this asx2+8x =(x+4)2 16In general:Boardworks Ltd 200512 of 50Completing the squareComplete the square for x2 10 x.Compare this expression to(x 5)2=x2 10 x+25=(x 5)2 25x2 10 x=x2 10 x+25 25 Complete the square for x2+3x.Compare this expression to Boardworks Ltd 200513 of 50Completing the squareHow can we complete the square for x2 8x+7?=(x 4)2 9x2 8x+7=x2 8x+16 16+7 Look at the coefficient of x.This is 8 so compare the expression to(x 4)2=x2 8x+16.In general:Boardworks Ltd 200514 of 50Completing the squareComplete the square for x2+12x 5.Compare this expression to(x+6)2=x2+12x+36=(x+6)2 41x2+12x 5=x2+12x+36 36 5Complete the square for x2 5x+7.Compare this expression to Boardworks Ltd 200515 of 50Completing the squareWhen the coefficient of x2 is not 1,quadratic equations in the form ax2+bx+c can be rewritten in the form a(x+p)2+q by completing the square.Complete the square for 2x2+8x+3.2x2+8x+3=2(x2+4x)+3By completing the square,x2+4x=(x+2)2 4 so2x2+8x+3=2(x+2)2 4)+3=2(x+2)2 8+3=2(x+2)2 5Take out the coefficient of x2 as a factor from the terms in x:Boardworks Ltd 200516 of 50Completing the squareComplete the square for 5+6x 3x2.By completing the square,x2 2x=(x 1)2 1 so5+6x 3x2=5 3(x 1)2 1)=5 3(x 1)2+3=8 3(x 1)2Take out the coefficient of x2 as a factor from the terms in x:5+6x 3x2=5 3(2x+x2)=5 3(x2 2x)Boardworks Ltd 200517 of 50Complete the square Boardworks Ltd 200518 of 50Contents Boardworks Ltd 200518 of 50Solving quadratic equationsQuadratic expressionsFactorizing quadraticsCompleting the squareSolving quadratic equationsThe discriminantGraphs of quadratic functionsExamination-style questions Boardworks Ltd 200519 of 50Quadratic equationsQuadratic equations can be solved by:completing the square,orfactorizationusing the quadratic formula.ax2+bx+c=0 (where a 0)The general form of a quadratic equation in x is:The solutions to a quadratic equation are called the roots of the equation.A quadratic equation may have:one repeated root,ortwo real distinct rootsno real roots.Boardworks Ltd 200520 of 50The roots of a quadratic equationIf we sketch the graph of a quadratic function y=ax2+bx+c the roots of the equation coincide with the points where the function cuts the x-axis.As can be seen here,this can happen twice,once or not at all.Boardworks Ltd 200521 of 50Solving quadratic equations by factorizationStart by rearranging the equation so that the terms are on the left-hand side:Factorizing the left-hand side gives usSolve the equation 5x2=3x5x2 3x=0 x(5x 3)=0or5x 3=05x=3x=0So Dont divide through by x!Boardworks Ltd 200522 of 50Solving quadratic equations by factorizationStart by rearranging the equation so that the terms are on the left-hand side.We need to find two integers that add together to make 5 and multiply together to make 4.Factorizing the left-hand side gives usSolve the equation x2 5x=4 by factorization.x2 5x+4=0(x 1)(x 4)=0 x 1=0orx 4=0 x=4Because 4 is positive and 5 is negative,both the integers must be negative.These are 1 and 4.x=1 Boardworks Ltd 200523 of 50Solving quadratics by completing the squareQuadratic equations that cannot be solved by factorization can be solved by completing the square.For example,the quadratic equationx2 4x 3=0can be solved by completing the square as follows:x=4.65x=0.646(to 3 s.f.)(x 2)2 7=0(x 2)2=7x 2=x=2+orx=2 Boardworks Ltd 200524 of 50Solving quadratics by completing the squareSolve the equation 2x2 4x+1=0 by completing the square.Write the answer to 3 significant figures.=2(x 1)2 1)+1=2(x 1)2 2+1=2(x 1)2 1Start by completing the square for 2x2 4x+1:2x2 4x+1=2(x2 2x)+1 Boardworks Ltd 200525 of 50Solving quadratics by completing the square2(x 1)2=12(x 1)2 1=0 x=1.71x=0.293 (to 3 s.f.)Now solving the equation 2x2 4x+1=0:or Boardworks Ltd 200526 of 50Using the quadratic equation formulaAny quadratic equation of the formcan be solved by substituting the values of a,b and c into the formulaax2+bx+c=0This formula can be derived by completing the square on the general form of the quadratic equation.Boardworks Ltd 200527 of 50Using the quadratic formulaUse the quadratic formula to solve 2x2+5x 1=0.2x2+5x 1=0 x=0.186x=2.69 (to 3 s.f.)or Boardworks Ltd 200528 of 50Using the quadratic formulaUse the quadratic formula to solve 9x2 12x+4=0.9x2 12x+4=0There is one repeated root:Boardworks Ltd 200529 of 50Equations that reduce to a quadratic formSome equations,although not quadratic,can be written in quadratic form by using a substitution.For example:Solve the equation t4 5t2+6=0.This is an example of a quartic equation in t.Lets substitute x for t2:x2 5x+6=0This gives us a quadratic equation that can be solved by factorization:(x 2)(x 3)=0 x=2Sot2=2t=or x=3t2=3t=or Boardworks Ltd 200530 of 50Contents Boardworks Ltd 200530 of 50The discriminantQuadratic expressionsFactorizing quadraticsCompleting the squareSolving quadratic equationsThe discriminantGraphs of quadratic functionsExamination-style questions Boardworks Ltd 200531 of 50The discriminantBy solving quadratic equations using the formula we can see that we can use the expression under the square root sign,b2 4ac,to decide how many roots there are.When b2 4ac 0,there are two real distinct roots.When b2 4ac=0,there is one repeated root:When b2 4ac 0,there are no real roots.Also,when b2 4ac is a perfect square,the roots of the equation will be rational and the quadratic will factorize.b2 4ac is called the discriminant of ax2+bx+c Boardworks Ltd 200532 of 50The discriminantWe can demonstrate each of these possibilities graphically.Boardworks Ltd 200533 of 50Contents Boardworks Ltd 200533 of 50Graphs of quadratic functionsQuadratic expressionsFactorizing quadraticsCompleting the squareSolving quadratic equationsThe discriminantGraphs of quadratic functionsExamination-style questions Boardworks Ltd 200534 of 50Plotting graphs of quadratic functionsPlot the graph of y=x2 4x+2 for 1 x 0(p,q)will be the minimum point.If a 0(p,q)will be the maximum point.Plotting the y-intercept,(0,c)will allow the curve to be sketched using symmetry.Boardworks Ltd 200546 of 50Exploring graphs of the form y=a(x+p)2+q Boardworks Ltd 200547 of 50Contents Boardworks Ltd 200547 of 50Examination-style questionsQuadratic expressionsFactorizing quadraticsCompleting the squareSolving quadratic equationsThe discriminantGraphs of quadratic functionsExamination-style questions Boardworks Ltd 200548 of 50Examination-style question a)Write 2x2 8x+7 in the form a(x+b)2+c.b)Write down the minimum value of f(x)=2x2 8x+7 and state the minimum value of x where this occurs.c)Solve the equation 2x2 8x+7=0 leaving your answer in surd form.d)Sketch the graph of y=2x2 8x+7.a)2x2 8x+7=2(x2 4x)+7=2(x 2)2 4)+7=2(x 2)2 8+7=2(x 2)2 1 Boardworks Ltd 200549 of 50Examination-style questionb)From this we can see that the minimum value of f(x)is 1.f(x)can be written as f(x)=2(x 2)2 1This occurs when x=2.c)2x2 8x+7=02(x 2)2 1=02(x 2)2=1(x 2)2=x 2=x=2 x=2 orx=2 Boardworks Ltd 200550 of 50Examination-style questiond)When y=0,x=2 or x=2+When x=0,y=7So the graph cuts the coordinate axes at(2+,0),(2 ,0)and(0,7).The parabola has a minimum at the point(2,1).yx712+2