Title: solve quad
11-4 Solving Quadratic Equations using Completing
the square and Quadratic formula
- A-REI.4a - complete the square to transform any
quadratic equation in x into (x-p)2 q that has
same solutions. - A-REI.4b - Solve, find the roots, the zeros, for
quadratic equations in 1 variable by sq rt,
factoring, quadratic formula - and completing the square.
Recognize when quadratic formula give complex
solutions, write correctly. - N-CN.7 --Solve quadratic equations with real
coefficients, and complex solutions - F-IF.8a.- Use the process of factoring and
completing the square in a quadratic function to
show zeros, extreme values, - and symmetry of the graph, and
interpret these in terms of a context.
2Completing the Square is based on Solving by
Square Roots
- Get the squared variable or quantity alone
- Square Root both sides
3Perfect Square Trinomials
- ax2 bx c 0
- a and b term are perfect squares
- 2ab b
- a2 2ab b2 (a b)2
- a2 2ab b2 (a b)2
4Solve these perfect square trinomials
This side is a perfect square trinomial
Rewrite it in its factored form
Now, square root both sides
Simplify
Solve for x
5Rewrite the expression as a perfect square
trinomial
- What would c have to be, to make these perfect
square trinomials?
6Complete the square, and show what the perfect
square trinomial is, then factor it
7If a quadratic equation is not a perfect square
trinomial, turning it into one is called
completing square
- 1. rewrite the equation into 1x2 bx c
- 2. complete the square, add (b/2)2 to both sides
of - 3. factor the perfect square trinomial
(x )2 c (b/2)2 - 4. square root both sides and solve for x
8Steps to solve by completing the square
- 1.) If the quadratic does not factor, move the
constant to the other side of the equation - Ex x²-4x -7 0 x²-4x7
- 2.) Work with the x² x side of the equation and
complete the square by taking ½ of the
coefficient of x and squaring - Ex. x² -4x 4/2 2²4
- 3.) Add the number you got to complete the square
to both sides of the equation - Ex x² -4x 4 7 4
- 4.)Simplify your trinomial square
- Ex (x-2)² 11
- 5.)Take the square root of both sides of the
equation - Ex x-2 v11
- 6.) solve for x
- Ex x2v11
9Solve by Completing the Square
10Solve by Completing the Square
11Solve by Completing the Square
12Solve by Completing the Square
13Solve by Completing the Square
14Solve by Completing the Square
15The coefficient of x2 must be 1
10
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16THE QUADRATIC FORMULA
- When you solve using completing the square on
the general formula you get - identify a, b, and c then substitute into the
formula and solve for x. - Best used when equation is not factorable
17WHY USE THE QUADRATIC FORMULA?
- The quadratic formula allows you to solve ANY
quadratic equation, even if you cannot factor it. - An important piece of the quadratic formula is
whats under the radical - b2 4ac
- This piece is called the discriminant.
-
18WHY IS THE DISCRIMINANT IMPORTANT?
- The discriminant tells you the number and types
of answers - (roots) you will get. The discriminant can be ,
, or 0 - which actually tells you a lot! Since the
discriminant is - under a radical, think about what it means if you
have a - positive or negative number or 0 under the
radical. -
19WHAT THE DISCRIMINANT TELLS YOU!
Value of the Discriminant Nature of the Solutions
Negative 2 imaginary solutions
Zero 1 Real Solution
Positive perfect square 2 Reals- Rational
Positive non-perfect square 2 Reals- Irrational
20Find the value of the discriminant and describe
the nature of the roots (real,imaginary,
rational, irrational) of each quadratic equation.
Then solve the equation using the quadratic
formula) 1.
a2, b7, c-11
Discriminant
Value of discriminant137 Positive-NON perfect
square Nature of the Roots 2 Reals - Irrational
Discriminant
21Example 1- continued
Solve using the Quadratic Formula
22Quadratic Formula (Example 2)
23- 3. 3x28x35
- 3x28x-350
- a3, b8, c -35
OR
24- 4. -2x2-2x3
- -2x22x-30
- a-2, b2, c -3