9.1 – Solving Quadratics by Square Roots Algebra 1 – Chapter 9 9.1 – Solving Quadratics by Square Roots Objective: Know terminology of square roots Find square roots Solve quadratic equations with square roots Evaluate using square roots Vocabulary: radical, radicand

Definition Square Root The square root of a number N is the number(s) that multiplied by itself equals N Positive numbers have two square roots Negative numbers have no square roots 0 has exactly one square root

Anatomy of a Square Root radical radicand

Find these square roots:

Solving Equations Using Square Roots We are allowed to take the square root of both sides of an equation…as long as both sides are POSITIVE! x2 = 4 x = 2 or -2

Steps for Solving Equations Using Square Roots NOTE: This only works if there is only x2 terms and constant terms! Move all x2 terms to the left side of the equation (either add or subtract). Move all constant terms to the right side (either add or subtract). Isolate the x2 (by dividing – make sure that x2 is positive). Take square root of both sides.

Solve these equations: 1. x2 = 25 2. x2 = 81 3. x2 = 7 4. x2 – 144 = 0 5. x2 + 12 = 0 6. x2 – 11 = 0 Move all x2 terms to the left side of the equation (either add or subtract). Move all constant terms to the right side (either add or subtract). Isolate the x2 (by dividing – make sure that x2 is positive). Take square root of both sides.

Solve these equations: Move all x2 terms to the left side of the equation (either add or subtract). Move all constant terms to the right side (either add or subtract). Isolate the x2 (by dividing – make sure that x2 is positive). Take square root of both sides. 7. 3x2 – 48 = 0 8. 12x2 – 60 = 0 9. 120 – 6x2 = -30

The Discriminant Part of the quadratic formula Tells us how many solutions a quadratic equation has This will be really important, but first we need to get good at evaluating it  ORDER OF OPERATIONS

Evaluate for When a = 1, b = -2, c = -3. Plug in a, b, and c. Simplify the radicand. Take square root.

Evaluate for 10. a = 16, b = -8, c = 1 11. a = 3, b = -4, c = 3 Plug in a, b, and c. Simplify the radicand. Take square root. 10. a = 16, b = -8, c = 1 11. a = 3, b = -4, c = 3

Falling Object Model h = -16t² + s h= height in feet When an object is dropped, the speed with which it falls continues to increase. Ignoring air resistance, its height h can be approximated by the falling object model. h = -16t² + s h= height in feet t = time in seconds s = initial height

Falling Object Model h = -16t² + s How long will it take a free-fall ride at an amusement park to drop 121 feet. Assume there is no air resistance. t = _____ seconds (Why not ±?)

If a bird egg fell out of a nest on the Sears Tower (1450ft), how long until it hits the ground? The Real World h = -16t2 + s

Brain Break Stay overnight A big “if” Take a few minutes to get up, stretch out, talk to a neighbor, or try the following rebus puzzles… Stay overnight A big “if”

L 9.1 Homework Tonight’s Homework P. 507-508 #’s 31-33, 39-41, 58-63, 83

Algebra T3 – Chapter 9 Daily Warm-Up Solve Each Equation. 2x² - 8 = 0 x² + 25 = 0

9.2 Simplifying Radicals Goals: 1. Use properties of radicals to simplify radicals. 2. Use quadratic equations to model real-life problems.

Product Property: The square root of a product equals the product of the square root of the factors.

Quotient Property: The square root of a quotient equals the quotient of the square root of the numerator and the denominator.

Find these square roots:

Simplify these square roots: Steps: Find the largest perfect square factor Split the radical Square root the perfect square PLEASE NOTE: You can also use a Factor Tree!

Simplify these square roots:

Simplify these square roots: Steps: Reduce the fraction Split the radical Square root perfect squares Look for perfect square factors and pull them out Reduce if possible

Simplify these square roots:

Simplify these square roots:

L 9.2 Homework Tonight’s Homework P. 514

Algebra T3 – Chapter 9 Daily Warm-Up Simplify the Expression.

9.1/9.2 Check Up  

9.3 Quadratic Functions Goal: 1. Sketch the graph of a quadratic function. 2. Use quadratic models in real life settings.

9.3 Graphing Quadratic Functions Objective: Identify a quadratic equation Identify the vertex and axis of symmetry of a parabola Graph a quadratic function Vocabulary: quadratic equation, standard form, parabola, vertex, axis of symmetry

Quadratic Equation and Function Quadratic Equation : polynomial equation with a degree of 2. It can be written in standard form as follows: ax² + bx + c = 0 Quadratic Function: a function that can be written in the standard form y = ax² + bx + c

Is each statement a quadratic function? If yes, find a, b, c. 1.) y = 4x2 + x – 4 2.) y = -13x2 – x 3.) y = x – 4 4.) y = 4x2 + 4 5.) y = -2x3 + 3x2 – x + 12

Graph of Quadratic Function Every quadratic function makes a u-shaped graph called a parabola. If the leading coefficient a is positive, the parabola opens up. If the leading coefficient a is negative, the parabola opens down. Example quadratic equation: y = -x² + 4

Graph of Quadratic Function The lowest or highest point of a parabola is called the vertex. The vertical line through the vertex that splits the parabola in half is called the axis of symmetry. Vertex Axis of Symmetry

Graphing of Quadratic Function Step 1: Find the x-coordinate of the vertex. x = -b 2a Step 2: Find the y-coordinate of the vertex, by substituting the x-coordinate into the equation. Step 3: Make a table of values, using x-values to the left and right of the vertex. Step 4: Plot the points and connect them with a smooth curve to form a parabola.

Graph y = 2x2 – 8x + 6 Step 3: Make a table x y 1 2 -2 3 4 Axis of symmetry is vertical line through the vertex. Same as x of vertex. x=2 Step 1: Find x of vertex a = and b= Step 2: Substitute in x of vertex to find y. Vertex is: ( , )

Now you try one! Graph y = -2x² - 4x + 5

Graph Analysis Graph y = x² – 2x – 3 Also list: Open up or down? Vertex? Axis of symmetry?

Check your info. x -1 1 2 3 y -3 -4 Opens Up Vertex is (1, -4). The axis of symmetry is x = 1 x -1 1 2 3 y -3 -4

9.3 Exit Questions Sketch a graph of the function. y = x² - 2x + 4 Does graph open up or down? Give coordinates of the vertex. Write the of the axis of symmetry.

L 9.3 Homework Tonight’s Homework P. 521-522 #’s 33-34, 46-49

9.4 Solving Quadratic Equations by Graphing Objective: Solve quadratic equations by graphing Solve Real World problems involving quadratic equations Vocabulary: zeros, roots

9.3 Check Up Graph: y = x2 + 2x Solve: 2x2 + 3 = 35 Solve: 2x2 + 3x = 35

Graph of Quadratic Function The lowest or highest point of a parabola is called the vertex. The vertical line through the vertex that splits the parabola in half is called the axis of symmetry. The x intercepts of a graph are called the solutions or zeros. We also call them roots! Solutions Vertex Axis of Symmetry

Solve by graphing 4x² = 16 -12 - 2 1 -12 -1 -2 x y 2 1 -12 -16 -1 -2 Put in Standard Form: 4x² - 16 = 0 a = b = c = Find vertex value = Chart x values and find intercepts -12 - 2 1 -12 -1 -2 x y 2 1 -12 -16 -1 -2 Solution: x = ± 2

Solve by graphing: x² - 4x = 5 Standard Form? Step 1: Find x of vertex Step 2: Substitute in x of vertex to find y. Step 3: Make a table of values until you find solutions. Step 4: Plot points to form parabola and state solutions.

Solve by graphing: x² - 4x = 5 y = x² - 4x – 5 Solution from graphing is x = 5, x = -1 What would factored equation be? y = (x – 5)(x + 1)

How can we solve a quadratic equation? Graphing: Put in standard form Fill out a table of values. Find x intercepts. Factoring: Put in standard form Factor by one of the four rules. Use zero product property to solve.

Now you try one! Solve y = 2x² - 4x – 6 by graphing Open up or down? Vertex? Axis of symmetry? Solutions? Factor to check?

x=1 (-1,0) (3,0) (1,-8)

9.4 Exit Questions 1. Solve by graphing: y = x² + 4x - 1 2. Solve by any method: 2x² = 32

9.4 Homework Tonight’s Homework P. 529-530 #’s 18, 28-29, 33-34

Algebra T3 – Chapter 9 Daily Warm-Up y = x² + 5x + 6 Solve the function by graphing. y = x² + 5x + 6

9.5 Solving Quadratic Equations by the Quadratic Formula Goal: 1. Use the quadratic formula to solve a quadratic equation and use quadratic models for real life situations.

The Quadratic Formula The solutions of the quadratic equation ax² + bx + c = 0 are When a ≠ 0 and b² - 4ac ≥ 0. You can read this formula as “x equals the opposite of b, plus or minus the square root of b squared minus 4ac, all divided by 2a.”

x² + 5x – 6 = 0 Step 1: Write is standard form x² + 5x – 6 = 0 Step 2: Identify a, b, c a = 1, b = 5, c = -6 Step 3: Plug into quadratic formula x² + 5x – 6 = 0 (1) (1)(-6) 5 x 2 4 - ± =

Solve 4x² - x – 7 = 0 Step 1: Write is standard form 4x² - x - 7 = 0 Step 2: Identify a, b, c a = 4, b = -1, c = -7 Step 3: Plug into quadratic formula

YOU TRY - Solve 2x² - 3x + 3= 0

YOU TRY - Solve 2x² - 3x – 8 = 0

Find the x intercepts of the graph of y = x² + 3x - 8 Finding the x intercepts is the SAME as finding the solutions to the equation. Use the Quadratic Formula to find the x intercepts

Vertical Motion Models Object is dropped: h = -16t² + s Object is thrown: h = -16t² + vt + s h = height (feet) t = time in motion (seconds) s = initial height (feet) v = initial velocity (feet per second) In these models the coefficient of t² is ½ the acceleration due to gravity. On the surface of Earth, this acceleration is about 32 feet per second per second. Why is there no middle term for the dropped object model?

From a 40 foot cliff, you throw a stone downward at 20ft/sec into the water below. How long will it take to hit the water?

L 9.5 Homework Tonight’s Homework P. 536-537 #’s 32-35, 55-56, 79

Algebra T3 – Chapter 9 Daily Warm-Up Find the x-intercepts of the graph of the Equation. y = x² - 11x + 24

Algebra T3 9.6 Applications of the Discriminant Objective: Use the discriminant to find the number of solutions to a quadratic equation Use the discriminant in real life

Discriminant Discriminant -b ± √ b² - 4ac 2a x =

Discriminant is b² - 4ac The discriminant can be used to find the number of solutions of the quadratic equation in the form of: ax² + bx +c 0 = x2 + 3x – 10 0 = x2 + 8x + 16 0 = x2 + 2x + 10 dis. is +, 2 real solutions dis. = 0, 1 solution dis. is -, no real solutions

How many solutions for: x² – 2x + 4 = 0 - b 4 ac (-2)² - 4(1)(4) = -12 Since answer is negative there is no real solution

How many solutions for: -3x2 + 5x = 1 Put in standard form: -3x² + 5x -1 a= -3, b = 5, c = -1 Use discriminant: b² - 4ac =

How many solutions for: -x2 = 10x + 25 Remember to put in standard form before identifying a, b, and c.

How many zeros for: y = x2 + 6x + 3 Remember the “zeros” are the x intercepts or solutions of the equation. This means there 2 zero’s or x intercepts.

y = x² + 6x + 3 Vertex x coordinate = -b/2a = -6/2(1) = -3 Y coord. = (-3)² + 6(-3) + 3 = 9 – 18 + 3 = -6 Vertex = (-3,-6) Note because a is + graph opens upward. This means there are 2 x intercepts

How many solutions for: y = x2 + 6x + 10 This means there are no zero’s or x intercepts.

y = x² + 6x + 10 Vertex x coordinate = -b/2a = -6/2(1) = -3 y coord. = (-3)² + 6(-3) + 10 = 9 – 18 + 10 = 1 Vertex = (-3, 1) Note because a is + graph opens upward. This means there are no x intercepts

How many zeros for: y = x2 + 6x + 9 This means there is one zero or x intercept.

y = x² + 6x + 9 Vertex x coordinate = -b/2a = -6/2(1) = -3 Ycoord. = (-3)² + 6(-3) + 3 = 9 – 18 + 9 = 0 Vertex = (-3,0) Note because a is + graph opens upward. This means there is 1 x intercept

YOU TRY - How many solutions for: -3x2 + 6x - 3 = 0 Remember to put in standard form before identifying a, b, and c.

L 9.6 Homework I am a protector. I sit on a bridge. One person can see right through me, while others wonder what I hide. What am I? Tonight’s Homework P. 544 #’s 9-17, *18-19

Algebra T3 – Chapter 9 Daily Warm-Up Sketch a graph of the function. y = x² + 5x + 6

9.7 Graphing Quadratic Inequalities Objective: Sketch the graph of a quadratic inequality.

9.3 - Graphing a Quadratic Function Step 1: Find the x-coordinate of the vertex. Step 2: Make a table of values, using x-values to the left and right of the vertex. Step 3: Plot the points and connect them with a smooth curve to form a parabola.

Sketching the Graph of a Quadratic Inequality Step 1) Replace inequality symbol(<,etc.) with = and sketch y = ax² + bx +c Dotted line for < or > Solid line for ≤ or ≥ Step 2) Pick a point clearly on inside or outside and test it in original inequality. Step 3) If test point is a solution shade its region. If not, shade the other region. ***Can also shade based on symbol and direction of parabola (Mental Math)

Sketch y ≥ ½x² + x Find Vertex Find Solution Dotted or Solid Test Point to Shade/Mental Shading

Sketch y ≥ x² + 2x

YOU TRY Sketch y ≤ -2x²

YOU TRY Sketch y > -x² - 2

L 9.7 Homework Tonight’s Homework P. 551-553 #’s 13-16, 25-28, 40 There is a frog lying dead in the center of a lily pad. The lily pad is sinking and there are two lily pads either side of him. Which lily pad should he jump to, the left or the right? Tonight’s Homework P. 551-553 #’s 13-16, 25-28, 40