Section 1.3 Quadratic Equations : 4 methods to solve them!
1. Zero­Factor Product Property
2. Square Root Property
3. Completing the Square
4. Quadratic Formula
Quadratic Equations: ax2+bx+c = 0 a, b, and c are real; a≠0.
• second degree (highest exponent is 2)
• graph is a parabola
• 2 solutions
Zero Product Property: If AB = 0, then A = 0 or B = 0 (or both)
A and B are factors.
Procedure:
1. Set the equation = 0
2. Factor
3. Set each factor = 0
4. Solve each equation for the variable
Examples:
Square Root Property: If A2 = c , then A = ±√c
Procedure:
1. Isolate the squared term on one side of the equation
2. Take the square root of both sides of the equation
3. Remember ± in order to find both solutions
4. Solve for the variable Examples:
Completing the Square: The goal is to create a perfect square trinomial that can be factored as ( )2. Once this is accomplished, you may use the Square Root Method to finish the problem.
Procedure:
1. Move the constant term "c" to the other side of the equation.
2. Divide each term on both sides of the equation by "a."
3. Add (b/2)2 to both sides of the equation. (thus completing the perfect square)
4. Factor the perfect square trinomial into ( )2.
5. Finish the problem by using the Square Root Method.
Examples:
Quadratic Formula: (can be derived using the Completing the Square Method!) MEMORIZE THIS FORMULA!!!!
1. Be sure the equation is set =0 so that ax2 + bx +c = 0
2. Use the coefficients in the formula and simplify:
x = ­b ±√b2 ­ 4ac
2a
3. Be sure to simplify twice for the ± part.
4. ***This formula applies to and works for ALL quadratic equations!
Example:
Using the Discriminant to determine number of and type of
solutions:
2 real solutions if:
1 real solution if:
2 imaginary solutions if:
When solving applications remember to set the quadratic equation = 0 before using the quadratic formula!
Set up: The cost, in dollars, of publishing x books is C(x) = 40,000 + 20x + .0001x2. How many books can be published for $250,000?
Gravity Problems: For objects in motion near the Earth, and neglecting other forces such as air resistance and friction, gravity pulls the object down to Earth. The height of the object is given by a quadratic equation:
h = ­ 1/2 gt2 + v0t + h0
h = height
g = force due to gravity (32 ft/sec2 or 9.8 m/sec2)
v0= initial velocity
h0= initial height
t = time (usually seconds)
Example: Robert stands on the topmost tier of seats in a baseball stadium, and throws a ball out onto the field with a vertical upward velocity of 60 ft/sec. The ball is 50 feet above the ground at the moment he releases the ball. When does the ball land?