Presentation on theme: "4.6 The Quadratic Formula and the Discriminant"— Presentation transcript:
1 4.6 The Quadratic Formula and the Discriminant Objectives:Solve quadratic equations by using the Quadratic FormulaUse the discriminant to determine the number and type of roots for a quadratic equation.
2 Quadratic FormulaAlways works to solve a quadratic equation, but is a little lengthy. The solutions of a quadratic equation of the form where a≠0 are given by the formula:
3 Examplex²-8x=33 x²-8x-33=0 Set = 0 a=1, b=-8, c=-33
4 Another Example7x²+6x+2=0 a=7, b=6, c=2 Since ALL of the coefficients are divisible by 2, simplify by dividing them by 2.
5 DiscriminantThe discriminant describes the solution to a quadratic equation. The part of the quadratic formula under the radical is the discriminant or b²-4ac.If b2 – 4ac > 0, and a perfect squareYou have two rational rootsIf b2 – 4ac >0, and not a perfect square.You have two irrational rootsIf b2 – 4ac = 0You have 1 real, rational root. (Repeated root)If b2 – 4ac < 0You have two complex roots
6 Examples b. 7x²-3x=0 (-3)²-4(7)(0)= 9-0=9 Two rational roots because 9 is positive and a perfect square.3x²-x+5=0(-1)²-4(3)(5)=1-60=-59Two complex roots because the discriminant is a negative.ExamplesFind the discriminant and describe the number and type of roots.x²-16x+64=0b²-4ac(-16)²-4(1)(64)==0One real, rational root because the discriminant equals 0.
7 We have discussed several methods for solving quadratic equations – which one do you use? Can Be UsedWhen to UseGraphingsometimesUse only if an exact number is not required. Best use to check the reasonableness of solutions found algebraicallyFactoringUse if the constant term is 0 or if the factors are easily determinedSquare Root PropertyUse for equations in which a perfect square is equal to a constantCompleting the SquarealwaysUseful for equations of the form ax2 + bx + c where b is evenQuadratic FormulaWhen other methods fail or are too tedious
8 Solve – use any method.4. x²+5x+8=0 Doesn’t factor, not easily done by completing the square (5 is odd) so use quadratic formula.1. 7x²-14x=07x(x-2)=0x=0, x=2x²-64=0x²=64x=8x²-16x+64=0(x-8)(x-8)=0x=8