6.6 Solving Quadratic Equations Objectives: Multiply binominals using the FOIL method. Factor Trinomials. Solve quadratic equations by factoring. Solve quadratic equations using the quadratic formula. Page 317
A binomial expression has just two terms (usually an x term and a constant). There is no equal sign. Its general form is ax + b, where a and b are real numbers and a ≠ 0. • One way to multiply two binomials is to use the FOIL method. FOIL stands for the pairs of terms that are multiplied: First, Outside, Inside, Last. • This method works best when the two binomials are in standard form (by descending exponent, ending with the constant term). • The resulting expression usually has four terms before it is simplified. Quite often, the two middle (from the Outside and Inside) terms can be combined.
The opposite of multiplying two binomials is to factor or break down a polynomial (many termed) expression. • Several methods for factoring are given in the text. Be persistent in factoring! It is normal to try several pairs of factors, looking for the right ones. • The more you work with factoring, the easier it will be to find the correct factors. • Also, if you check your work by using the FOIL method, it is virtually impossible to get a factoring problem wrong. • Remember! When factoring, always take out any factor that is common to all the terms first.
A quadratic equation involves a single variable with exponents no higher than 2. • Its general form is where a, b, and c are real numbers and . • For a quadratic equation it is possible to have two unique solutions, two repeated solutions (the same number twice), or no real solutions. • The solutions may be rational or irrational numbers.
To solve a quadratic equation, if it is factorable: • 1. Make sure the equation is in the general form. • 2. Factor the equation. • 3. Set each factor to zero. • 4. Solve each simple linear equation.
To solve a quadratic equation if you can’t factor the equation: • Make sure the equation is in the general form. • Identify a, b, and c. • Substitute a, b, and c into the quadratic formula: • Simplify.
The beauty of the quadratic formula is that it works on any quadratic equation when put in the form general form. • If you are having trouble factoring a problem, the quadratic formula might be quicker. • Always be sure and check your solution in the original quadratic equation.
Factor x2 - 7x + 12. 1. Pairs of numbers which make 12 when multiplied: (1, 12), (2, 6), and (3, 4). • 1 + 12≠7. 2 + 6≠7. 3 + 4 = 7. Thus, d = 3 and e = 4. • (x - 3)(x - 4) • Check: (x - 3)(x - 4) = x2 -4x - 3x + 12 = x2 - 7x + 12 • Thus, x2 - 7x + 12 = (x - 3)(x - 4).
Factor 2x3 +4x2 + 2x. First, remove common factors: 2x3 +4x2 +2x = 2x(x2 + 2x + 1) • Pairs of numbers which make 1 when multiplied: (1, 1). • 1 + 1 = 2. Thus, d = 1 and e = 1. • 2x(x + 1)(x + 1) (don't forget the common factor!) • Check: 2x(x + 1)(x + 1) = 2x(x2 +2x + 1) = 2x3 +4x2 + 2x • Thus, 2x3 +4x2 +2x = 2x(x + 1)(x + 1) = 2x(x + 1)2.x2 + 2x + 1 is a perfect square trinomial.
Homework Assignment on the Internet Section 6.6 (Read Solving Quadratic Equation) Pp 329-330: 2-78even.