Section 2.2 part 1 Solving Quadratic Equations Dr. Joseph Kolacinski Assistant Professor Elmira College

Quadratic Equations:

Quadratic Equations Definition : A quadratic equation is an equation which can be written in the form: ax 2 + bx + c = 0, where a is not zero. To solve a quadratic equation, we break it down into two simpler equations that we already know how to solve. The fact that allow us to do this is called the “Zero-Factor Property.”

The Zero-Factor Property:

The Zero-Factor Property If the product of any two real numbers is zero, then at least one of the numbers must be zero. i.e. If ab = 0 then a = 0 or b = 0.

Solving by Factoring::

Solving by Factoring: The easiest way to solve one of these equations is by factoring. To solve a quadratic equation by factoring you: Move all the terms to one side of the equation so that it is in the form “ ax 2 + bx + c = 0. ” Factor the non-zero side of the equation. Set each of the two factors equal to zero (by the Zero-Factor Property). Solve the resulting equations to get the solution or solutions.

An Example:

An Example Solve: 2x 2 + 5x = 3 1. Move all the terms to one side of the equation so that it is in the form “ ax 2 + bx + c = 0. ” 2x 2 + 5x – 3 = 0 2. Factor the non-zero side of the equation. (2x – 1)(x + 3) = 0 3. Set each of the two factors equal to zero (by the Zero-Factor Property). 2x – 1 = 0 or x + 3 = 0 4. Solve the resulting equations to get the solution or solutions. x = ½ or x = – 3

Examples:

Examples Solve each of the following quadratic equations by factoring.

Solving by Extracting Roots:

Solving by Extracting Roots If a quadratic function is written in the form for some positive real number n , then u must be one of the square roots of n , that is: Using this notion to solve a quadratic equation is called “extracting (square) roots.”

Example of Extracting Roots:

Example of Extracting Roots Solve ( x + 2) 2 = 25 Take the square root of both sides and solve for x .

More Examples:

More Examples Solve each of the following quadratic equations by extracting roots.

Completing the Square:

Completing the Square We’ve seen that if a quadratic equation is written in the form then u must be one of the square roots of n : The main importance of the technique of “extracting roots” is as the basis of the technique of “completing the square.”

The Main Idea:

The Main Idea A ² + 2 AB + B ² = ( A + B )² A ² – 2 AB + B ² = ( A – B )² In other words, if our equation looks like this: ( x + 2) 2 = 25 We can solve it by extracting roots. To make this happen we want one side of our equation to be a perfect square trinomial. That is to say, one of these:

How to think about it::

How to think about it: A ² + 2 AB + B ² = ( A + B )² x 2 + 6 x + ? ² = ( x + ? )² We’ll illustrate on an example. Say we want to solve x 2 + 6 x – 10 = 0 by completing the square. We can add ten to both sides of the equation to get it out of our way. x 2 + 6 x = 10 Then we have to ask ourselves, what do we have to add to both sides of the equation so that we have a perfect square on the left? Compare our left-hand-side to a perfect square.

What to add?:

What to add? We’re really asking, what’s B? If we line stuff up, it’s clear that A = x. but then 2 B = 6, which means B = 3. In other words, we want to add 9 to both sides. Now we can finish: A ² + 2 AB + B ² = ( A + B )² x 2 + 6 x + ? ² = ( x + ? )²

Finishing Our Example.:

Finishing Our Example. x 2 + 6 x = 10 x 2 + 6 x + 9 = 10 + 9 ( x + 3) 2 = 19 In practice, as long as the coefficient of x 2 is 1, we can take half the coefficient of x, square it, and add that to both sides.

Thinking Geometrically.:

Thinking Geometrically. x 2 + 6 x x 2 + 6 x + 9 =( x + 3) 2 Say we want to complete the square for… To get both sides of our square equal, we can cut our rectangle in half and reposition it. x 2 + 6 x Then, to finish the big square, we need to add a small square that’s 3 by 3. That’s half the width of our original rectangle on each side. The result is a perfect square trinomial.

Completing the Square:

Completing the Square To solve a quadratic equation by completing the square we: Move the constant term away from the variable terms. If necessary, divide both sides of the equation by the coefficient of x 2 . This makes your leading coefficient 1. Take half the coefficient of x, square it and add that quantity to both sides of the equation. Factor your polynomial using a perfect square trinomial formula. A ² + 2 AB + B ² = ( A + B )² or A ² – 2 AB + B ² = ( A – B )² Solve the resulting equation by extracting roots.

An Example::

An Example: Solve the following equation by completing the square: Move the constant term away from the variable terms. If necessary, divide both sides of the equation by the coefficient of x 2 . This makes your leading coefficient 1.

An Example::

An Example: So far, we have: Take half the coefficient of x, square it and add that quantity to both sides of the equation.

An Example::

An Example: So far, we have: Factor your polynomial using a perfect square trinomial formula. Solve the resulting equation by extracting roots.

More Examples::

More Examples: Solve the following equations by completing the square:

What if you can’t factor?:

What if you can’t factor? Factoring should be our first strategy when we want to find the solution to a quadratic equation. We’ve seen that, if you can’t factor, it still may be possible to find real number solutions to a quadratic equation. ax 2 + bx + c = 0 The two tools that we have so far are extracting roots and completing the square. We’d like something more general and/or shorter. If we complete the square on a generic quadratic equation, we’ll derive a usable formula.

Completing the Square for a Generic Quadratic Equation:

Remember the steps: Move the constant term away from the variable terms. Completing the Square for a Generic Quadratic Equation Solve by Completing the Square If necessary, divide both side of the equation by the coefficient of x 2 . This makes your leading coefficient one.

Completing the Square for the General Quadratic Equation (cont’d):

Completing the Square for the General Quadratic Equation (cont’d) So far we have: Take half the coefficient of x , square it and add that quantity to both sides of the equation.

Completing the Square for the General Quadratic Equation (cont’d):

Completing the Square for the General Quadratic Equation (cont’d) So far we have: Factor your polynomial using a perfect square trinomial formula. Solve the resulting equation by extracting roots.

Completing the Square for the General Quadratic Equation (end):

Completing the Square for the General Quadratic Equation (end) So far we have: Finally This is The Quadratic Formula that gives the solution to the general quadratic equation

So, if you can’t factor…:

So, if you can’t factor… If you can’t factor, if there are solutions to a quadratic equation: they will be given by the quadratic formula:

An example::

An example: Use the quadratic formula to find the solutions to: x 2 – 6x + 4 = 0 Notice that here we have a = 1, b = – 6 and c = 4.

Two tips::

Two tips: In general, it is a bad idea to try to use the quadratic formula unless you cannot factor. You should always try to solve by factoring first. Secondly, the quadratic formula needs to be in your brain so that you can use it easily. Memorize it first, then do the homework. That way, doing the homework will reinforce your memory.

Examples::

Examples: Try to solve each of the following quadratic equations by factoring. If factoring is not possible, solve using the quadratic formula.

How Many Solutions?:

How Many Solutions? Some quadratic equations don’t have real number solutions. For example, 2x 2 + x + 3 = 0 has no real solutions. We can see this because, if we substitute a = 2, b = 1 and c = 3 into the quadratic formula we get: Which can’t be a real number, since the expression under the radical is negative.

The Discriminant:

The Discriminant The piece of the quadratic formula, is called the “discriminant” of the quadratic equation. It tells us how many real number solutions a quadratic equation has. that lives under the radical, namely

Finding the Number of Solutions:

Finding the Number of Solutions Sign # of real solutions 2 real solutions 1 rational solution No real solutions The number of real solutions the quadratic equation: Can be found using the discriminant as follows:

Example:

The discriminant is positive, so there are 2 real unequal roots, which are: Example Find the number of real solutions, of the quadratic equation and then solve it. 4 x 2 − x − 2 = 0

Examples::

Examples: Calculate the discriminant of each of the following equations, then determine how many real solutions each equation has.

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