The Quadratic Formula

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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!

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
In this exercise, radicals are not given in simplest form.