WARM UP 1. Simplify 2. Multiply 3. Divide

QUADRATIC EQUATIONS INTRODUCTION

 Solve equations of the type ax + bx + c =0 OBJECTIVES  Solve a quadratic equation by completing the square.  Solve problems using quadratic equations

 The second-degree or quadratic equation models projectile motion by relating the time t an object is in the air to the initial velocity and the height h of an object. INTRODUCTION DEFINITION An equation of the type where a, b, and c are constants and a ≠ 0, is called standard form of a quadratic equation.

STANDARD FORM EQUATIONS  Every quadratic polynomial with complex coefficients can be factored into two linear factors. Example Solve This is an equation of the type where c = 0. Quadratic equations in standard form where c = 0 can be easily solved by factoring. Factoring or Using the principle of zero products These numbers check, so the solutions are 0 and. A quadratic equation of this type will always have 0 as one solution. Sometimes it helps to find standard form before factoring.

MORE EXAMPLES  Solve (x – 1)(x + 1) = 5(x – 1) Multiplying or Finding standard form These numbers check, so the solutions are 4 and 1. Factoring Using the principle of zero products

Solve TRY THIS…

MORE EXAMPLES  Consider any quadratic equation in standard form where b = 0, that is, an equation of the form. We can use the multiplication and addition principles to obtain an equation of the form where or Solve Divide both sides by 1/3 or Using the addition property We can abbreviate this as. Sometimes we get solutions that are complex numbers.

MORE EXAMPLES Solve Divide both sides by 1/4 or Adding 9 to both sides Finding square roots or

Solve TRY THIS…

COMPLETING THE SQUARE  The trinomial is the square of a binomial, because. Given the first two terms of a trinomial, we can find the third term that will make it a square. This process is called completing the square. Example: Complete the square for What must be added to to make it a trinomial square? We take half of the coefficient of x and square it. Half of 12 is 6, and 6 is 36. We add 36. is a trinomial square. It is equal to.

MORE EXAMPLES  Complete the square for Thus, is a trinomial square. It is Half of ¾ (the coefficient of y) is 3/8.

Complete the square TRY THIS…

MORE EXAMPLES  We can solve quadratic equations of the form by completing the square. Adding 5 to both sides Solve by completing the square: Adding 1 to complete the square or The numbers check, so they are the solutions. The solutions can be abbreviated as

Solve by completing the square TRY THIS…

MORE EXAMPLES  For many quadratic equations the leading coefficient is not 1, but we can use the multiplication principle to make it 1. Multiplying both sides by ¼ Solve by completing the square: Take ½ of 3, then add = 9/4. or

Solve by completing the square TRY THIS…

CH. 8.1 HOMEWORK Textbook pg. 345 #2, 6, 10, 14, 22, 24, & 26