Ideas

Ideas

Ideas

**Solving Quadratic Equations by Factoring**

Objectives

Solve quadratic equations by factoring

State, illustrate, and apply the zero – product property of real numbers in solving quadratic equations.

InfoMath

Although in Latin the prefix "quadri" means four, the word "quadrus" means a square (because it has four sides) and "quadratus" means "squared." Consider other words such as: "quadrille," meaning a square dance; "quadrature," meaning constructing a square of a certain area.

Quadratic equations were originally used in connection with geometric problems involving squares. Since the second power is called a "square," equations with the highest power being two are called quadratic equations.

Let's Do this!

Group yourself into four groups. Each group will complete the following table using the given expressions.

Evaluation:

Solve for the roots of the following quadratic equations

1. x exp 2 + 3x – 40 = 0

2. a exp 2 + 10a + 21 = 0

3. 8x exp 2 – 2x = 3

4. 8x exp 2 – 57x + 54 = 0

Generalization

In solving for the roots of quadratic equation, we simply factor the trinomial and square each binomial factor to zero to obtain the value of the unknown variable or roots of the quadratic equations.

Factoring as a tool in solving equations is limited, as you might have noticed. It can only be used in those quadratic equations where the quantity ax2 + bx + c is factorable.

**Demonstration Lesson**

**The National Teachers Colleges The School of Advanced Studies Teaching in Mathematics**

Definition: quadratic equations by factoring are normally expressed as where a does not equal zero.

Mathematical Ideas

Information

Types of Special products Polynomials Factors

Difference of two squares

Common Monomial Factor

Perfect Square Trinomial

Quadratic Trinomial

1. 3x exp 2 – 5x +2 (2x + 3)exp 2

2. X exp 2 – 25 x( x – 9)

3. X exp 2 – 9x (x + 5)(x – 5)

4. 4x exp 2 + 12x + 9 (3x -2) (x – 1)

Zero – Product Property of Real Numbers.

If a and b are real numbers, then ab = 0 if and only if a = 0 or b = 0.

To solve a quadratic equation by factoring:

1. Start with the equation in the standard form. Be sure it is set equal to zero!

2. Factor the left hand side (assuming zero is on the right)

3. Set each factor equal to zero

4. Solve to determine the roots (the values of x)

Factoring with DOTS

(difference of two squares)

Or Isolate the Variable

(Square Root Property)

Factoring with GCF

(greatest common factor)

Find the largest value that can be factored from each of the elements of the expression.

Factoring Trinomials

In a quadratic equation in descending order with a leading coefficient of one, look for the product of the roots to be the constant tern and the sum of the roots to be the coefficient of the middle term.

Factoring Harder Trinomials

If the leading coefficient is not equal to 1, you must think more carefully about how to set up your factors.

Tricky One!!

Be sure to get the equation set equal to zero before you factor.

More Examples

The factors of x exp 2 + 3x - 4 are:

(x+4) and (x-1)

Why? Well, let us multiply them to see:

(x+4)(x-1) = x(x-1) + 4(x-1)

= x exp 2 - x + 4x - 4

= x exp 2 + 3x - 4

Multiplying (x+4)(x-1) together is called Expanding.

Example: 6x exp 2 + 5x - 6

Example: 2x exp 2 + 7x + 3

Assignment:

Solve for the roots of the following quadratic equations

1. x exp 2 + 5x + 6 = 0

2. x exp 2 – 4x + 77 = 0

3. x exp 2 – 49 = 0

4. 9x exp 2 = 144

**Group 4**

**"Best Group"**