Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Solving Quadratic Equations

Solving Quadratic Equations

A quadratic equation is an equation of the form,

ax2 + bx + c = 0 ,

where a, b, and c are real numbers, with a ≠ 0 . The condition, a ≠ 0 , ensures that the
equation actually does have a x2-term. When solving quadratic equations, we consider
two cases: b = 0 and b ≠ 0 .

When b = 0, a quadratic equation is the form of ax2 − c = 0 so we use the square
root property to quadratic equations when b = 0.

Square Root Property

For any real number k, the equation x2 = k is equivalent to .

If k > 0, then x2 = k has 2 real solutions.

k < 0, then x2 = k has no real solution.

k = 0, then 0 is the only solution to x2 = k .

Examples: Solving ax2 − c = 0

Solutions:

(a) Use the square root property to solve x2 − 9 = 0 .



Thus, the solution set for the equation x2 − 9 = 0 is {-3, 3}.

(b) Using the square root property to solve, we get

Thus, the solution set for the equation is .

(c) Use the square root property to solve .



Because the square of any real number is nonnegative, the equation
has no real solution.

For the case when b ≠ 0 , we can solve quadratic equations by factoring,
completing the square, and using the quadratic formula.

Solving Quadratic Equations by Factoring

Zero Factor Property

If A and B are algebraic expressions, then the equation AB=0 is equivalent to the
compound statement A = 0 or B = 0.

Examples:

Solve the following quadratic equations by factoring.

(a) x2 − x −12 = 0

(b) (x + 3)(x − 4) = 8

Solutions:

Using the zero factor property, we get

Thus, the solution set is {-3, 4} .

(b) (x + 3)(x − 4) = 8

First, multiply the left side using the FOIL method and subtract 8 from both sides.

Using the zero factor property, we get

Thus, the solution set is {-4, 5} .

Solving Quadratic Equations by Completing the Square

To complete the square of x2 + kx , add to both sides. That is, add the
square of half the coefficient of x to both sides.

Examples:

Solve the following quadratic equations using completing the square

(a) x2 + 6x + 7 = 0

(b) 2x2 −3x − 4 = 0

Solutions:
(a) First, subtract 7 from both sides and then add to both sides:

The solution set is .

(b) First, divide both sides by the leading coefficient, which is 2.

Now, add 2 to both sides and to both sides.

Thus, the solution set is .

Solving Quadratic Equations using the Quadratic Formula

Quadratic Formula

The solution to ax2 + bx + c = 0 , with a ≠ 0 , is given by the formula,

provided b2 − 4ac ≥ 0 . If b2 − 4ac < 0 , there are NO REAL SOLUTIONS.

Example: Solve the quadratic equation, x2 + 8x + 6 = 0 , using the quadratic
formula.

Solution: Since a = 1, b = 8, and c = 6 ,

Thus, the solution set is .