Quadratic Equations and Expressions are very important on the GMAT. Firsty you will be tested on these concepts in Data Sufficiency Problems and Word Problems. Secondly they are the polynomials (higher than degree one) which come on the GMAT. You will encounter them in many inequalities problems and therefore, the concept of graphs and curves pertaining to quadratic expressions is very important.
introduction
A quadratic equation in one unknown is an equation of the form: 
, where 
.
A quadratic equation may have two real roots, one double root or no real roots.
If 
and 
, the roots of this equation is given by
.
This formula can be used to solve all quadratic equations in the form 
with 
.
Solve. Solution. We put
= 1,
= –2 and
= 2 to the quadratic formula. Then we have
is not a real number and it is defined that
.
This is not a solution on the GMAT!!
Solve.
Solution. Let, then the original equation becomes
Since
for real
, the solution
is rejected. Then we have
, which gives
Solve.
Solution. Note that the equation is equivalent to
.
Let, then the equation becomes
Sum and product of roots
Let 
and 
be the roots of a quadratic equation 
with 
, then
(1) 
and
(2) 
.
Proof. From the quadratic formula, we have
and
.
Alternatively, we can find these two relations by comparing coefficients. Since 
and 
are roots to the equation, we have
.
The original equation is 
and by comparing coefficients, we have
and
.
- Note that the relation of sum and product of roots with the coefficients always holds, no matter if the roots are real or not.
Letand
be the roots of
and
.
Find the value of–
.
Solution.
Note that
Since
, we have
.
Discriminant and nature of roots
As we can see from the quadratic formula, the term inside the square root sign, i.e. 
, determines the nature of roots. This term is usually denoted as 
and is called the discriminant. The value of the discriminant tells us the number of real roots of the quadratic equation.
For a quadratic equation 
, where 
, and its discriminant is 
,
(1) if 
0, the equation has two distinct real roots.
(2) if 
= 0, the equation has one double real root.
(3) if 
0, the equation has no real root, it has two distinct unreal roots.
Find the number of real root(s) for (1)
(2)
(3)
Solution. We first calculate the value of the discriminant of the equation and then we can determine the number of real root(s).
For (1),, thus there are two distinct real roots.
For (2),, thus there is no real root.
For (3),, thus there is one real root.
Find the range of values ofsuch that
has real root(s).
Solution. For the equation to have real root(s),.
Thus we have
Quadratic functions and their graphs
A quadratic function is a function in the form 
, where 
. The graph of a quadratic function is a parabola. If 
0, the parabola opens upward; if 
0, the parabola opens downward. These two cases are shown in Figure 1 and Figure 2 respectively.
Figure 1: The graph of 
, where 0
Figure 2: The graph of 
, where 
0
Maxima and Minima
For a parabola, it has a vertex and an axis of symmetry. The coordinates of the former and the equation of the latter can be found from 
by completing the square. ![f (x) = ax2 + bx + c ( 2 -b c) [ = a x + a( x +) a ( ) ] = a x2 + 2 ⋅ b-⋅ x + b- 2 +-c- b- 2 [(2a ) 2a a ] 2a = a x + b- 2 + 4ac-b2- 2a 2a](images/quadratic_Quadratic_Equations_and_Functions92x.png)
If 
is positive, since 
for all real 
, 
and 
, 
( 
) attains its minimum when 
.
If 
is negative, since 
, this gives 
and hence 
( 
) attains its maximum when 
.
For both cases, the vertex is 
and the axis of symmetry is 
.
The vertex corresponds to the maximum or minimum value of the function.
Find the maximum value or minimum value, if exist, ofand
.
Solution.
Thus the minimum value of
(
) is 5.
Thus the maximum value of(
) is
.
![g(x) = (- 2x2 + 3)x - 1 = - 2 x2 - 3 x - 1 [ ( 2)2] ( )2 = - 2 x2 - 32x + 12 ⋅ 32 - 1 + 2 12 ⋅ 32 ( )2 = - 2 x + 34 + 18](images/quadratic_Quadratic_Equations_and_Functions114x.png)
Inequalities
The graph of a quadratic function may cut the 
-axis at two points, one point or it may not intersect with the -axis, depending on the sign of the discriminant. For 
, the value of the discriminant 
tells us the number of real roots of the equation 
, i.e. the number of intersection point(s) of the graph of 
( ) and = 0.
Case I : If 
0 and 
0, the equation 
has two real roots and thus the parabola of 
cuts the 
-axis at two distinct points.
Either side of the bowl, the curve is in the +ve Y-axis.
Case II :If 
= 0 and 0, the parabola touches the 
-axis, for 
has only one real root.
The curve lies entirely in the +ve Y-axis, except at 
, where 
Case III :If 
0 and 0, 
has no real root and hence the graph of 
( 
) does not intersect with the 
-axis. The whole parabola lies above the 
-axis.
The curve lies entirely in the +ve Y-axis. So the quadratic expression is positive for all real values of x
Case IV :If 0 and 
0 (inverted), has two real root and hence the graph of ( ) intersects with the -axis at two distinct points.
Case V :If = 0 and 0 (inverted), has one real root (equal roots) and hence the graph of ( ) intersects with the -axis at only one distinct.
The curve lies entirely in the –ve Y-axis, except at 
, where 
Case VI : If 0 and < 0 (inverted), has no real root and hence the graph of ( ) does not intersect with the -axis. The whole parabola lies below the -axis
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