Cornell Notes Quadratic Equations

DEFINITION OF A QUADRATIC EQUATION

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

ax bx c2 0,

where a, b, and c are real numbers, with a ≠ 0.

A quadratic equation in x is also called a second-degree polynomial equation in x.

THE ZERO-PRODUCT PRINCIPLE

To solve a quadratic equation by factoring, we apply the zero-product principle which states that:

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.

SOLVING A QUADRATIC EQUATION BY FACTORING

2. Factor completely.

3. Apply the zero-product principle, setting each factor containing a variable equal to zero.

4. Solve the equations in step 3.

5. Check the solutions in the original equation.

EXAMPLE: SOLVING QUADRATIC

EQUATIONS BY FACTORING (1 OF 3)

Solve by factoring: 2x2 x 1.

Step 1 Move all nonzero terms to one side and obtain zero on the other side.

2x x2 1 0

Step 2 Factor

(2x− 1)(x + 1) = 0

EQUATIONS BY FACTORING (2 OF 3)

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

(2x− 1)(x + 1) = 0

EXAMPLE: SOLVING QUADRATIC

EQUATIONS BY FACTORING (3 OF 3)

Step 5 Check the solutions in the original equation.

2x x2 1

Check

EXAMPLE: SOLVING QUADRATIC

EXAMPLE: SOLVING QUADRATIC EQUATIONS BY FACTORING

• Solution

• Step 1- Move all nonzero terms to one side and obtain zero on the other.

• Step 2 Factor- 2x(2x-1)=0

• Step 3 and 4- Set each factor equal to zero and solve the resulting equations.

2x=0 or 2x-1=0 x=0

2x=1

x=1/2

• Step 5

B) SOLUTION

CHECK POINT SOLVE BY FACTORING

SOLVING QUADRATIC EQUATIONS BY THE SQUARE ROOT PROPERTY

EXAMPLE: SOLVING QUADRATIC EQUATIONS BY THE SQUARE ROOT PROPERTY

Solve by the square root property:

SOLUTION

SOLUTION

SOLUTION

CHECK POINT: SOLVE BY THE SQUARE ROOT PROPERTY:

COMPLETING THE SQUARE

EXAMPLE: CREATING PERFECT SQUARE TRINOMIALS BY COMPLETING THE SQUARE

CHECK POINT: SOLVING A QUADRATIC EQUATION BY COMPLETING THE SQUARE

EXAMPLE: SOLVING A QUADRATIC EQUATION BY COMPLETING THE SQUARE

CHECK POINT: SOLVE BY COMPLETING THE SQUARE.

THE QUADRATIC FORMULA

EXAMPLE: SOLVING A QUADRATIC EQUATION USING THE QUADRATIC

FORMULA (1 OF 2) Solve using the quadratic formula:

2x2 2x 1 0

a = 2, b = 2, c = −1

EXAMPLE: SOLVING A QUADRATIC EQUATION USING THE QUADRATIC FORMULA (2 OF 2)

CHECK POINT:

THE DISCRIMINANT

The discriminant of the quadratic equation determines the number and type of solutions.

THE DISCRIMINANT AND THE KINDS OF SOLUTIONS TO A X SQUARED + B X + C = 0

• If the discriminant is positive, there will be two unequal real solutions.

• If the discriminant is zero, there is one real (repeated) solution.

• If the discriminant is negative, there are two imaginary solutions.

EXAMPLE: USING THE DISCRIMINANT