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THE QUADRATIC FORMULA

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The graphs of quadratic functions can have three different x-intercept situations, as shown below:
no x-intercepts;
exactly one x-intercept;
two different x-intercepts.

no x-intercepts exactly one x-intercept two different x-intercepts

Points on the x-axis have their y-value equal to zero.
Thus, to find the x-intercepts for any curve, you set  y  equal to zero and solve for  x .

In particular, to find the x-intercepts of a quadratic function  y=ax2 +bx+c  (a&neq;0),
it is necessary to solve the equation   ax2 +bx+c=0  .
The formula that gives the solutions to this equation is called the quadratic formula, and is derived next:

DERIVATION OF THE QUADRATIC FORMULA:
solving the equation  ax2 +bx+c=0   (a&neq;0)

Using the technique of completing the square,
the equation is transformed to the form   z2 =k :
ax2 +bx+c=0 (original equation)
ax2 +bx=-c (subtract  c  from both sides)
x2 +ba x=- ca (divide both sides by a&neq;0;
the technique of completing the square only works
when the coefficient of the squared term is 1)
x2 + bax + ( b2a ) 2 =- ca + b2 4a2 (add  ( b2a ) 2 =b2 4a2  to both sides;
this is the correct number to add
to turn the left side into a perfect square)
(x+ b2a )2= -ca ⋅4a 4a+ b2 4a2 (rewrite left side as a perfect square;
get a common denominator on the right)
(x+ b2a )2= b2 -4ac4 a2 (add fractions, simplify)
x+b 2a=± b 2-4ac 4a2 (the equation  z2 =k  has solutions  z=± k )
x+b 2a=±   b 2-4ac 2|a | (use the rules  A B= A B ,   AB= A⋅ B ,    and   A 2=| A| )
Notice that if  a>0 , then  |a|=a , so the right side becomes  ±  b 2-4ac 2a .
Also, if  a<0 , then  |a|=-a , so the right side becomes  ∓  b 2-4ac 2a .
In both cases, the same two values for the right side result.
Thus, we continue with a simplified right side:

x+b 2a=±  b 2-4ac 2a (the plus/minus allows elimination of the absolute value)
x=- b2a ±  b2 -4ac 2a (add  -b 2a  to both sides)
x=-b ±b 2-4ac 2a (add fractions)

In summary, we have:

THE QUADRATIC FORMULA
The solutions to the equation  ax2 +bx+c=0   (a&neq;0)  are given by:

x=-b ±b 2-4ac 2a


The expression  b2 -4ac  that appears under the square root in the quadratic formula
is critical in determining the nature of the solutions to the quadratic equation  ax2 +bx+c=0  , as follows:

If  b2 -4ac>0 , then there are two different solutions to the quadratic equation,
and hence the quadratic function  ax2 +bx+c  has two different x-intercepts.

If  b2 -4ac=0 , then there is exactly one solution to the quadratic equation,
and hence the quadratic function  ax2 +bx+c  has only one x-intercept.

If  b2 -4ac<0 , then there are no real number solutions to the quadratic equation,
and hence the quadratic function  ax2 +bx+c  has no x-intercepts.

Thus, the expression  b2 -4ac  helps us to discriminate between
the various types of solutions to a quadratic equation,
and the various x-intercept situations for a quadratic function.

Thus, we have the following definition:

DEFINITION:  discriminant

The expression  b2 -4ac  is called the discriminant of the quadratic equation  ax2 +bx+c=0   (a&neq;0) .

Similarly, the expression  b2 -4ac  is called the discriminant of the quadratic function  ax2 +bx+c   (a&neq;0) .


Click here to explore the relationship between the discriminant and x-intercepts, with Geometer's Sketchpad!



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Algebra II Table of Contents

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