The Quadratic Formula
9-4: The Quadratic Formula Standard 19.0 Students know the quadratic formula and are familiar with its proof by completing the square. Standard 20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. Standard 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points
To find any solution…
Why use it? Graphing and Factoring can only find roots that are INTEGERS. Completing the Square can find any root. The Quadratic Formula will find ANY root, works on really ugly numbers, AND can tell us right away if we will have 0, 1, or 2 roots!
The Quadratic Formula takes more memorization, but less thinking… Just Plug and Chug!
Quadratic Formula Proof by Completing the Square!
Solving by Factoring A. Solve x2 – 2x – 35 = 0. Round to the nearest tenth if necessary. Method 1 Factoring x2 – 2x – 35 = 0 (x –7)(x + 5) = 0 x –7 = 0 or x + 5 = 0 x=7 x = –5
Original equation Factor x2 –2x – 35. Zero Product Property Solve for x.
Same Equation with Quadratic Formula Method 2
Quadratic Formula Quadratic Formula
a = 1, b = –2, and c = –35 Multiply.
Continued… Add. Simplify.
Separate the solutions.
or
=7
= –5
Answer: The solution set is {–5, 7}.
B. Solve 15x2 – 8x = 4. Round to the nearest tenth if necessary.
Step 1
Rewrite the equation in standard form.
15x2 – 8x = 4 15x2 – 8x – 4 = 4 – 4 15x2 – 8x – 4 = 0 Step 2
Original equation Subtract 4 from each side. Simplify.
Apply the Quadratic Formula. a = 15, b = –8, and c = –4
Multiply.
or
Add.
Separate the solutions.
Practice 1. Solve x2 + x – 30 = 0. 2. Solve 20x2 – 4x = 8. 3. x2 – 2x – 24 = 0 4. 24x2 – 14x = 6 5. x2 + 3x - 18 = 0 6. 4x2 + 2x = 17
SPACE TRAVEL Two possible future destinations of astronauts are the planet Mars and a moon of the planet Jupiter, Europa. The gravitational acceleration on Mars is about 3.7 meters per second squared. On Europa, it is only 1.3 meters per second squared. Using the equation given, find how much longer baseballs thrown on Mars and on Europa will stay above the ground than similarly thrown baseballs on Earth.
Baseball Thrown on Mars
Baseball Thrown on Europa
These equations cannot be factored, and completing the square would involve a lot of computation.
To find accurate solutions, use the Quadratic Formula.
Since a negative number of seconds is not reasonable, use the positive solutions.
SPACE TRAVEL The gravitational acceleration on Venus is about 8.9 meters per second squared, and on Callisto, one of Jupiter’s moons, it is 1.2 meters per second squared. Suppose a baseball is thrown on Callisto with an upward velocity of 10 meters per second from two meters above the ground. Find how much longer the ball will stay in air than a similarly-thrown ball on Venus. Use the equation
where H is the height of an object
t seconds after it is thrown upward, v is the initial velocity, g is the gravitational pull, and h is the initial height.
about 14.5 seconds
HW: p. 497 9 – 22 ALL
To find any solution…
Why use it? Graphing and Factoring can only find roots that are INTEGERS. Completing the Square can find any root. The Quadratic Formula will find ANY root, works on really ugly numbers, AND can tell us right away if we will have 0, 1, or 2 roots!
The Quadratic Formula takes more memorization, but less thinking… Just Plug and Chug!
Quadratic Formula Proof by Completing the Square!
Solving by Factoring A. Solve x2 – 2x – 35 = 0. Round to the nearest tenth if necessary. Method 1 Factoring x2 – 2x – 35 = 0 (x –7)(x + 5) = 0 x –7 = 0 or x + 5 = 0 x=7 x = –5
Original equation Factor x2 –2x – 35. Zero Product Property Solve for x.
Same Equation with Quadratic Formula Method 2
Quadratic Formula Quadratic Formula
a = 1, b = –2, and c = –35 Multiply.
Continued… Add. Simplify.
Separate the solutions.
or
=7
= –5
Answer: The solution set is {–5, 7}.
B. Solve 15x2 – 8x = 4. Round to the nearest tenth if necessary.
Step 1
Rewrite the equation in standard form.
15x2 – 8x = 4 15x2 – 8x – 4 = 4 – 4 15x2 – 8x – 4 = 0 Step 2
Original equation Subtract 4 from each side. Simplify.
Apply the Quadratic Formula. a = 15, b = –8, and c = –4
Multiply.
or
Add.
Separate the solutions.
Practice 1. Solve x2 + x – 30 = 0. 2. Solve 20x2 – 4x = 8. 3. x2 – 2x – 24 = 0 4. 24x2 – 14x = 6 5. x2 + 3x - 18 = 0 6. 4x2 + 2x = 17
SPACE TRAVEL Two possible future destinations of astronauts are the planet Mars and a moon of the planet Jupiter, Europa. The gravitational acceleration on Mars is about 3.7 meters per second squared. On Europa, it is only 1.3 meters per second squared. Using the equation given, find how much longer baseballs thrown on Mars and on Europa will stay above the ground than similarly thrown baseballs on Earth.
Baseball Thrown on Mars
Baseball Thrown on Europa
These equations cannot be factored, and completing the square would involve a lot of computation.
To find accurate solutions, use the Quadratic Formula.
Since a negative number of seconds is not reasonable, use the positive solutions.
SPACE TRAVEL The gravitational acceleration on Venus is about 8.9 meters per second squared, and on Callisto, one of Jupiter’s moons, it is 1.2 meters per second squared. Suppose a baseball is thrown on Callisto with an upward velocity of 10 meters per second from two meters above the ground. Find how much longer the ball will stay in air than a similarly-thrown ball on Venus. Use the equation
where H is the height of an object
t seconds after it is thrown upward, v is the initial velocity, g is the gravitational pull, and h is the initial height.
about 14.5 seconds
HW: p. 497 9 – 22 ALL
