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: 2×2 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:
2×2 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
