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Chapter 9 Quadratic Equations And Functions By Chris Posey and Chris Bell 9-1 Square Roots You already know how to find the square of a number i.e. 4²=16 To find the square root, you basically do the opposite of squaring a number. i.e. √(16)=4 Apply Skills Learned • Now that you know how find the square root, let’s try some problems for practice! • Find the square roots of the following problems. 3. √(400) 3. √400=20 2. √(16) 4 √(36) 4. √36=6 2. √16=4 1. √(144) 1. √122=12 9-2 Solving by Quadratic Equations by Finding Square Roots Example 1> x2=4 x= +√4 or - √4 x= 2 or -2 Example 2> 3x2-48=0 3x2=48 x2=16 x= √16 x=4 or -4 Write the original equation Find square roots 22=4 and (-22)=4 Write the original equation Add 48 to each side Divide each side by 3 Find square roots 42=16 and (-4)2=16 Apply Skills Learned • Now that you know how to solve quadratic equations by finding square roots, find the answers to the following problems. 2. x2=225 3. x2-15=10 3. x=5 2. x=15 1.5t2-125=0 1. t=5 9.3 Simplifying Radicals • Product Property of Radicals: √ab= √a ∙ √b Example> √50= √(25∙2)= 5 √2 Example> √48= √(4∙12)= 4 √3 • Quotient Property of Radicals: √a/b= √a/√b Example> √32/50= √(2∙16)/√(2∙25) Factor using square roots =√(16/25) =√(16)/√(25) =4/5 Divide common factors Use Quotient Property Simplify Apply Skills Learned Now that you know how to simplify radicals, try it out on your own! 1.√(9/49) 2. √(18) 3. √(196) 1)3/7 2)3√2 3)14 9.4 Graphing Quadratic Functions • A quadratic function is a function that can be written as a formula Y=ax2+bx+c, where a≠0 This will give a graph with a u shape called a parabola. If a is grater than 0, it opens up. If it is negative, then it opens down. Graphing (9.4) • Find the x coordinate of the vertex, which is x=-b/2a • Make a x,y table and use the x values • Plot the points and connect them with a smooth curve to form the parabola Example • Sketch graph of y=x2-2x-3 • Find x coordinate of vertex. -b/2a=-2/2(1)=1 • Make a table x| -2 -1 0 1 2 3 4 y|5 0 -3 -4 -3 0 5 (1,-4) is the vertex, plot the rest of the points and draw a curve connecting them. Practice • Find vertex coordinates, and make a x,y value table using x values to the right and left of the vertex. • y=-4x2-4x+8 y|-16 0 8 9 8 0 -16 (-1/2,9) x|-3 -2 -1 -.5 0 1 2 9.5 Solving Quadratic Equations by Graphing • Solutions for the quadratic graphs are the x-axis intercepts, where y=0. • This number can be checked in the original equation, by setting it equal to 0. Example •y=x2-2x-3 is shown here. Note that the x intercepts are located at -1 and 3. •If substituted for x, the equation would result in zero, the solutions for the equation. Practice • Solve the equation algebraically, check your answers by graphing. • 2x2+8=16 Solutions are ±2 9.6 Solving Quadratic Equations by the Quadratic Formula • The solutions of the quadratic equation, ax2+by+c=0, are (-b+/-√(b2-4ac) x= --------------------(2a) when a≠0 and b2-4ac≥0 Example • Solve x2+9x=14=0 • Solution 1x+9x+14=0 (-b+/-√(b2-4ac) x= --------------------(2a) (-(9)+/-√((9)2-4(1)(14)) x= ------------------------------(2(1)) -9+/-√(25) x=-------------------2 -9+/-5 X=---------------2 There are 2 solutions x=-2, and x=-7 Practice • Solve the quadratic equation. • 2x2-3x=8 X=2.89, and x=-1.39 9.7 Using the Discriminant • The discriminant is the radical expression in the quadratic formula ie. (-b+/-√(b2-4ac)) x= --------------------(2a) If the discriminant is positive, then the solution has 2 solutions. If it is zero, it has one solution. If it is negative, there are no real solutions. Example • Find value of the discriminat and determine if it has two solutions, one solution, or no solutions. • x2-3x-4=0 • Use the equation, ax2+bx+c=0 to identify values, ie. a=1, b=-3, c=-4 • Substitute into the discriminant b2-4ac=(-3)2-4(1)(-4) =9+16 =25 Discriminant is positive, therefore two solutions. Practice • Determine whether the graph will intersect the x-axis at one, two, or zero points. • y=x2-2x+4 It does not intersect the axis 9.8 Graphing Quadratic Inequalities • The graph of a quadratic inequality consists of all the points (x,y) that are part of the inequality. Quadratic inequalities can be represented by; y> or < or or ax2+bx+c • When graphing, use a dashed line for the parabola when the equality is > or< • Use a solid line when it is or • The parabola separates the graph into two sections. A test point is a point that is not on the graph. If the test point is a solution, then shade the region, if not shade the other region. Example • Solve –x2 + 4 < 0. • Find the x-axis intercepts –x2 + 4 = 0 x2 – 4 = 0 (x + 2)(x – 2) = 0 x = –2 or x = 2 • Use the origional inequality to find the area to shade • y<0, therefore shade everything outside the parabola, below the x-axis. Practice • Determine whether the orderd pair is a solution of the inequality. • y≤x2+7, (4,31) Point is a solution outside of the parabola Now remember to study hard, because we’ll be watching you….