1.5 Quadratic Equations

Start p 145 graph and model for #131 & discuss.

Definition of a Quadratic Equation

A quadratic equation in x is an equation that can be written in the standard form

ax2 bx c 0

where a, b, and c are real numbers with a not equal to 0. A quadratic equation in x is also called a second-degree polynomial equation in x.

The Zero-Product Principle

If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero.

If AB 0, then A 0 or B 0.

Q: Will this work for any other number, such as AB=5?

Solving a Quadratic Equation by Factoring

1. If necessary, rewrite the equation in the form ax2 bx c 0, moving all terms to one side, thereby obtaining ______ on the other side.

2. Factor.3. Set each factor = zero. (Apply the

zeroproduct principle.)4. Solve the equations in step 3.5. Check the solutions in the __________

equation.

Text Example

• Solve 2x2 7x 4 by factoring and then using the zeroproduct principle. (Do not look at notes, no need to write.)

Step 1 Move all terms to one side and obtain zero on the other side. Subtract 4 from both sides and write the equation in standard form.

2x2 7x 4 4 42x2 7x 4 Step 2 Factor.

Solution cont.

• Solve 2x2 7x 4 by factoring and then using the zeroproduct principle.

Steps 3 and 4 Set each factor equal to zero and solve each resulting equation.

2 x 1 or x 4 2 x 1 x 4x = 1/2

Steps 5 check your solution (by putting each solution back into the ORIGINAL equation to see if it yields a TRUE statement.

Ex: Solve for x:(2x + -3)(2x + 1) = 5 Why can’t we set each factor =5?

ALWAYS begin by

factoring out the GCF.

Q: In the above example, it is not necessary to set the factor 4 = 0, but what if the GCF had been 4x?

Simplify

Set = 0

Factor

Apply zero product principle

Check.

The Square Root Method

d

If u is an algebraic expression and d is a positive real number, then u2 = d has exactly two solutions.If u2 = d, then u = or u = -Equivalently,If u2 = d then u =

We only use this method if the variable is originally contained within a “squared part”. Ex: x2-8=12 or (2x-4)2 –5=20. Can you think of a counter example? Do: d

d

25(4 1) 12x

Text Example

What term should be added to the binomial x2 + 8x so that it becomes a perfect square trinomial? Then write and factor the trinomial.

x2 + 8x +____2

= (x + ____)2

Note: this is still an expression, not an equation.Do (factor by completing the square- see instructions next

slide first) p 144 # 54.2 6 5 0x x

Completing the Square

If x2 + bx is a binomial, then by adding (b/2) 2, which is the square of half the coefficient of x, a perfect square trinomial will result. That is,

x2 + bx + (b/2)2 = (x + b/2)2

That is, take half of the coefficient of the x term, square it, and add it to each side.

Then take +/- the square root of each side. (Square root method.)

Note: this is really just using the fact that the square of a binomial results in a perfect square trinomial. We are just “completing” the perfect square trinomial.

Solving by the Quadratic Formula

a

acbbx

2

42

Given a quadratic equation in the form:

02 cbxaxWe can solve for x by “plugging in” a, b and c:

Derived by completing the square, if interested, see p121.

a>0,

a,b,c integers

2 5 8x x Ex: Solve by using the quadratic formula:

Put into form

Identify a,b,c

Plug in

Simplify

Common errors:

Not writing the division bar all the way.

-b means – (whatever b is!). In this case –(-8) = 8.

No x-intercepts

No real solution; two complex imaginary solutions

b2 – 4ac < 0

One x-intercept

One real solution (a repeated solution)

b2 – 4ac = 0

Two x-intercepts

Two unequal real solutionsb2 – 4ac > 0

Graph of y = ax2 + bx + c

Kinds of solutions to ax2 + bx + c = 0

Discriminantb2 – 4ac

The Discriminant and the Kinds of Solutions to ax2 + bx +c = 0

Which approach do we use to solve a quadratic equation?

1. Recognize that you have a quadratic equation.2. If the variable is isolated within the “squared part”,

isolate the squared part, take +/- square root of each side, then isolate the variable. (Square root method.)

3. Otherwise set = 0a. If it is EASY to factor, factor, set each factor equal to zero and solve for the variable (Factoring method.)b. If it is NOT easy to factor, plug a, b, and c into the quadratic formula and simplify (Quadratic formula method.)

4. If it says to solve by completing the square, do so (Completing the square method.)

The Pythagorean TheoremThe sum of the squares of the lengths of the

legs of a right triangle equals the square of the length of the hypotenuse.

If the legs have lengths a and b, and the hypotenuse has length c, then

a2 + b2 = c2 do # p144:105, 138 (set up)