Presentation on theme: "Exploring Quadratic Functions and Inequalities"— Presentation transcript:
1 Exploring Quadratic Functions and Inequalities Advanced AlgebraChapter 6
2 Solving Quadratic Functions Solve the following equation.Square ofa BinomialSolution:
3 Solving Quadratic Functions Multiply the following expressions.Is there a pattern?Shortcut Method( x ) = x x2×product of both terms1st termlast termsquare of1st termsquare oflast term
4 Solving Quadratic Functions Try using the shortcut method with these.Now Try Backwards:x2 + 8x = ( )2x2 – 4x + 4 = ( )2x2 + x + ¼ = ( )2THINK!!!x + ½
5 Solving Quadratic Functions by Completing the Square For example, solve the following equation by completing the square.Step 1 Move the constant to the other side.Step 2 Notice the coefficient of the linear term is 3, or b = 3. Therefore, is the new constant needed to create a Square Binomial. Add this value to both sides.
6 Solving Quadratic Functions by Completing the Square Step 3 Factor and Solve.
7 Quadratic FormulaAnother way to solve quadratic equations is to use the quadratic formula.This is derived from the standard form of the equation ax2 + bx + c = 0 by the process of completing the square.
8 Quadratic Formula The Quadratic Formula The value of the discriminant, b2 – 4ac, determines the nature of the roots of a quadratic equation.The Discriminant
9 Discriminant b2 – 4ac Value Description Sample Graph b2 – 4ac is a perfectsquareb2 – 4ac = 0b2 – 4ac < 0b2 – 4ac > 0Intersects the x-axis once.One real root.Does not intersect the x-axis.Two imaginary roots.Intersects the x-axis twice.Two real, irrational roots.Intersects the x-axis twice.Two real, rational roots.
10 Solving Quadratic Functions with the Quadratic Formula For example, solve the following equation with the quadratic formula.Step 1 Write quadratic equation in Standard Form.Step 2 Substitute coefficients into quadratic formula. In this case a = 4, b = –20 and c = 25The discriminant, (–20)2 – 4(4)(25) = 0.There is one real, rational root.
11 Solving Quadratic Functions with the Quadratic Formula For example, solve the following equation with the quadratic formula.Step 1 Write quadratic equation in Standard Form.Step 2 Substitute coefficients into quadratic formula. In this case a = 3, b = –5 and c = 2The discriminant, (–5)2 – 4(3)(2) = 1.There are two real, rational roots.