Solving Quadratic Equations Lesson Essential Question: How do you solve quadratic equations? What does the solution(s) mean or represent?

Solving Quadratic Equations

What does it mean to solve a quadratic equation???? The "solutions" to the Quadratic Equation are where it is equal to zero. Where: ax2 + bx + c = 0 Where the graph crosses or touches the x-axis. They are also called "roots", or sometimes "zeros"

Let’s take a look at where the solutions are and how many there can be!

Basketball Shot Revisited We used the function: h(t) = 8 + 40.8t – 16t2 to represent the height of the basketball as a function of the time the basketball was in the air after it was released from the players’ hands 8 feet above the ground.

Basketball Shot Revisited To find the time when the shot would reach the 10 foot height of the basket, you would solve the equation: 8 + 40.8t – 16t2 = 10 To find the time when an “air ball” would hit the floor, you would solve the equation: 8 + 40.8t – 16t2 = 0

Basketball Shot Revisited 8 + 40.8t – 16t2 = 10 and 8 + 40.8t – 16t2 = 0 The values of t that satisfy the equations are called the solutions of the equations. For each situation, you could get good estimates of the solutions by searching in tables of values or by analyzing the graph of the function. Let’s do that now. Let’s take a look at the graph and table of the function.

Let’s Start Getting Exact!!! LEQ: What are some effective methods for solving quadratic equations algebraically to get exact solutions?

Do You Know??? x2 = 36 What is the solution?

Some quadratic equations can be solved by use of the fact that for any positive number n, the equation x2 = n is satisfied by two numbers:

Solving Quadratics Solving a quadratic equation is similar to solving an equation You want to get x all by itself, so you need to UNDO the equation. There may only be a squared term and a constant. (an ax2 and c term) Add or subtract the constant to its like term on the other side of equals if needed. Divide the a term on both sides of the equals. Take the square root of both sides of the equals. Declare your solution as both a positive and negative value. Round to the nearest tenth if your solutions are a decimal. You cannot take the square root of a negative #, write no solution for this kind of problem.

4x2 = 100

2x2 + 6 = 18

5x2 + 40 = 20

-5x2 + 40 = 20

Practice the Concept Solve the quadratic equations contained on the worksheet, “Solving Simple Quadratic Equations”. In each case, you will check your reasoning by substituting the proposed solution values for x into the original equation.

Platform High Diver If the diver jumps off a 50-foot platform, what rule gives her or his distance fallen d (in feet) as a function of time t (in seconds)? Think really, really, really hard…you’ve seen this equation before!!!!

Platform High Diver Equation: 16t2 = 20 The rule: d = 16t2 Write and solve an equation to find the time required for the diver to fall 20 feet. Equation: 16t2 = 20

Can I have a volunteer to show their work and solution on the board? Equation: 16t2 = 20 Determine the solution(s) to the quadratic equation. 16t2 = 20 Can I have a volunteer to show their work and solution on the board?

Platform High Diver Function: h(t) = 50 - 16t2 What function gives the height, h, of the diver at any time, t, after she or he jumps from the 50-foot platform ? Function: h(t) = 50 - 16t2

Platform High Diver Equation: 50 - 16t2 = 0 Write and solve an equation to find the time when the diver hits the water. Equation: 50 - 16t2 = 0

Can I have a volunteer to show their work and solution on the board? Equation: 50 - 16t2 = 0 Determine the solution(s) to the quadratic equation. 50 - 16t2 = 0 Can I have a volunteer to show their work and solution on the board?

Now let’s move onto solving quadratic functions by factoring. Factoring Quadratics Now let’s move onto solving quadratic functions by factoring.

Soccer Player h(t) = –16t2 + 24t If a soccer player kicks the ball from a spot on the ground with initial upward velocity of 24 ft/s, the height of the ball h (in feet) at any time, t, seconds after the kick will be approximated by the quadratic function: h(t) = –16t2 + 24t

Soccer Player 0 = –16t2 + 24t What time will the ball hit the ground? Think about what your height is on the ground? We must set the h(t) equal to zero because that is the height from the ground. 0 = –16t2 + 24t

Soccer Player 0 = 24t – 16t2 Be Careful!!!! This equation has to be solved differently than the square root method because it has the “bx” term in it!! We must use our GCF skills!!

Soccer Player 0 = 24t – 16t2 Let’s factor 24t – 16t2. Factoring with GCF: 0 = 8t(3 – 2t) Now we have to solve for t!!

Soccer Player 0 = 24t – 16t2 The expression 8t(3 – 2t) will equal 0 when 8t = 0 and when 3 - 2t = 0. Why?

Soccer Player h(t) = 24t – 16t2 To find the solutions of the equation 24t – 16t2 = 0 we need to solve the factors!! Solve 8t = 0 Solve 3 - 2t = 0.

Let’s Practice! Solve the following quadratic equations: 3x2 – 6x = 0 Let’s verify our answers by looking at the graph of the function. 3x = 0 and x – 2 = 0 x = 0 and x = 2 4x2 + 2x = 0 2x(2x +1) = 0 Let’s verify our answers by looking at the graph of the function. 2x = 0 and 2x + 1 = 0 x = 0 and x = - ½

Let’s Practice Again! Solve the following quadratic equations: 7x2 + 14x = 0 9x2 – 36 = 0 7x2 + 14x = 0 7x(x + 2) = 0 Let’s verify our answers by looking at the graph of the function. 7x = 0 and x + 2 = 0 x = 0 and x = -2 9x2 – 36 = 0 9(x2 – 4) = 0 Let’s verify our answers by looking at the graph of the function. 9 = 0 and x2 – 4 = 0 x = 2 and x = -2

Let’s Practice Once More! Solve the following quadratic equations: -x2 – 10x = 0 -4x2 – 12x = 0 -x2 – 10x = 0 -x(x + 10) = 0 Let’s verify our answers by looking at the graph of the function. -x = 0 and x + 10 = 0 x = 0 and x = -10 -4x2 – 12x = 0 -4x(x + 3) = 0 Let’s verify our answers by looking at the graph of the function. -4x = 0 and x + 3 = 0 x = 0 and x = -3

Let’s Practice Once More! Solve the following quadratic equations: x2 – 25 = 0 16x2 – 576 = 0 x2 – 25 = 0 x2 = 25 Let’s verify our answers by looking at the graph of the function. x = 5 and x = -5 16x2 – 576 = 0 16x2 = 576 Let’s verify our answers by looking at the graph of the function. x2 = 36 x = 6 and x = -6

Practice the Concept Solve the quadratic equations contained on the worksheet, “Solving Quadratic Equations by factoring out GCF”.

Check For Understanding Complete the Check For Understanding worksheet.

Ticket Out the Door Complete the Ticket Out the Door assignment and turn it in on your way out the door please.

Homework Assignment Complete the “Solving Quadratic Equations Experts Now” worksheet for homework please.