Based on the CAT tests from the past few years, there are usually two or three questions. The problems in this chapter are not all that hard to figure out. Since this part is about the high school curriculum, it is important for tests like JMET, XAT, SNAP, etc. In this chapter, we look at quadratic equations, inequations, and polynomial formulas with a higher degree of degree.
Definition of the Standard Quadratic Expression
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If a \(\neq\) 0, the phrase ax 2 + bx + c is termed the quadratic expression for the real numbers a, b, and c. Here, a is the x2 coefficient, b is the x coefficient, and c is a constant term.
All of the following functions are quadratic functions since the degree of each equation is 2. That means in each function there is always a term x2 or ax2 for every non-zero value of a. Here x is a variable and a, b, c are the constant numbers.
y = x2
y = ax2
y = ax2 + x
Properties of the Quadratic Equations
- The degree of any quadratic equation is always 2.
- The value of x that satisfies the relation ax2+bx+c = 0 is called the root or zero or solution of this equation.
- A quadratic equation has exactly two roots (or solutions or zeros).
- quadratic equation cannot have more than two different roots.
- A quadratic equation can have either two or zero REAL roots.
- If α is a root of the quadratic equation ax2+bx+c = 0, then (x − α) is a factor of ax2 +bx+c = 0
- If α and β are the two roots of the quadratic equation ax2+bx+c = 0 ,then
ax2+bx+c = 0 can be expressed as the product of two factors (x −α) and (x − β) as following
ax2 +bx+c = (x−α)(x−β) = 0
- If (x −α) and (x − β) are the two factors of the quadratic equation ax2 + bx + c = 0, then (x − α )(x − β) = 0(x −α) = 0 \(\implies\) x = α or (x −β) = 0 \(\implies\) x = β
Where α and β are the roots of the equation ax2 + bx + c = 0
- If quadratic equation is satisfied by more than two distinct numbers (real or imaginary) then it becomes an identity i.e., a = b = c = 0.
Becoming an identity implies that whatever value you substitute for x, the equation will be satisfied for every value of x.
Characteristics of the Graph of the Quadratic Function
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- A quadratic equation graph is exactly a parabola, as seen below.
- A Parabola are always symmetric.
- The "axis of symmetry" is the line that cuts the parabola across the center.
- The axis of symmetry is given by x = \(\frac{b}{2a}\). It denotes the distance between the axis of symmetry and the Y – axis.
- Depending on the quadratic function, the axis of symmetry may or may not overlap the Y-axis. However, the axis of symmetry is always parallel to the Y – axis.
- The "vertex" is the point on the axis of symmetry that bisects the parabola and is the point with the greatest curvature.
- The parabola can only open UP or DOWN.
- If the parabola expands up, the vertex will be at the lowest point of the graph; if it opens down, the vertex will be at the highest point.
- The graph's lowest point is called the minimum (plural is minima), and its greatest point is called the maximum (plural is maxima).
- Obtain the vertex of the quadratic graph by y = \(\frac{4ac-b^2}{4a}\)
Different Ways to Express the Quadratic Equation
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A quadratic equation can be expressed in the following forms
- ax2 + bx + c = 0
- a2 + \((\frac{b}{a})x + (\frac{c}{a}) = 0\)
- x2 − (α+β)x + (αβ) = 0; where α and β are the roots of the equation.
Solutions or Roots of the Quadratic Equations
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The value of x that satisfies the relation ax2 +bx+c = 0is called the root or solution of this equation. Simply, the root of the quadratic function y = ax2 +bx+c = 0 is the value of x for which y becomes 0.
Sum and Product of the Roots of a Quadratic Equation
If the two roots of the quadratic equation ax2+bx+c = 0 are α and β,
Sum of roots (α + β) = \(-\frac{b}{a}\)
Product of the roots (αβ) = \(\frac{c}{a}\)
Methods of Solving the Quadratic Equation
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(1) Factorization Method
(2) Sridharacharya Method
(3) Square Completion Method
(4) Sum and Product of Roots Method
Nature of the Roots of the Quadratic Equation
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For a quadratic equation ax2 +bx+c = 0; where a,b,c are real numbers and a ≠ 0, the nature of the roots can be easily determined by knowing the value of the discriminant (D = b2 − 4ac) of the above quadratic equation.
D < 0 – Complex Non-zero imaginary parts Unequal (Conjugate pairs)
D = 0 – Real Rational & Equal
D > 0 (D is a perfect square) – Real Rational & Unequal
D > 0 (D is not a perfect square) – Real Irrational & Unequal (Conjugate Pairs)
- Roots are real only when D is non-negative.
- Roots are complex (or imaginary) when D is negative.
- Roots are rational only when D is a perfect square number like 0, 1, 4, 9, 16, ...... etc.
- Roots are equal only when D =0.
- The equal roots are called Double Root.
- When the roots are irrational they are in conjugate pairs as if one root is p + \(\sqrt q\) , the other root will be p − \(\sqrt q\) . Here, q is the irrational part of the root.
- When the roots are complex they are in conjugate pairs as if one root is p+iq ,the other root will be p−iq. Here iq is the imaginary part of the root.
- If D > 0; a = 1; b,c ∈ Z (integer numbers) and roots are rational, the roots are integers.
- If a quadratic equation has one real root and a, b, c∈R, other root is also real.
- If the roots are real and equal, the graph will touch the X -axis.
- If the roots are real and unequal, the graph will intercept the X -
- axis.
- If the roots are non-real, the graph does not touch the X -axis.
- The points on the X -axis, where the quadratic graph touches or intercepts are called the x-intercepts.
Maximum or Minimum Value of a Quadratic Equation
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A quadratic equation, as you know, when drawn on a plane takes the shape of a parabola. This parabolic graph opens only in two directions –
(a) Upward (like hands-up) when a person is happy.
(b) Downward (like hands-down) when a person is sad.
These graphs can be used to describe scenarios in which a person is pleased when his attitude an is positive and unhappy when his attitude an is negative. So, in the quadratic equation y = ax2 +bx+c, the coefficient of x2 is positive, i.e., a. The parabolic graph rises upward, offers the smallest value of y. Similarly, when the coefficient of x2 in the quadratic equation y = ax2 + bx + c is negative, the parabolic graph expands downward, offers the maximum value of y.
Graphically, the maxima (or minima) denote the maximum (or minimum) distance between X -axis and the vertex of the graph.
Previous year questions on Quadratic equation
Ques 1: If p and q are the roots of the equation x2 − bx + c = 0, then what is the equation if the roots are ( pq + p + q) and ( pq − p − q) ? (CAT 2016)
- x2 −2cx+ (c2 −b2) = 0
- x2 −2bx+ (b2 + c2) = 0
- cx2 −2(b+ c)x+ c2 = 0
- x2 + 2bx−(c2 −b2) = 0
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Solution: (A)
Sum of roots = p + q = b ……….. (i)
Product of roots = pq = c ……….. (ii)
Formulating quadratic equation with roots (pq + p + q) and (pq − p − q).
Sum of roots = pq + p + q + pq – p – q = 2pq
From Eq (ii)
pq = c
Sum of roots = 2c
Product of roots = (pq + p + q) (pq − p – q)
= (pq)2 − (p + q)2
From Eqs. (i) and (ii), we get Product of roots = c2 − b2
∴ Required equation is x2 − 2cx + c2 − b2 = 0.
Ques 2: If x4 − 8x3 + ax2 − bx + 16 = 0 has positive real roots, then find a − b. (CAT 2016)
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Solution:
Let p, q, r and s be the roots of the equation
p+q+r+s=8
pqrs = 16
This happens only when
p = q = r = s = 2
p+q+r+s)/4 = 2
\(\sqrt[4]{pqrs} = 2\)
Arithmetic mean is equal to geometric mean .
This is possible only when all the numbers are equal.
p = q = r = s = 2
pq+ pr+ ps+ qr+ qs+ rs=a
24 = a
pqr + pqs+ prs+ qrs= b
32 = b
a − b = 24 − 32 = − 8
Ques 3: Let u and v be the roots of the equation x2 − 2x + p = 0 and let y and z be the roots of the equation x − 18x + q = 0. If u < v < y < z are in arithmetic progression, then p, q respectively equal to (CAT 2013)
- 8,17
- 3,7
- - 3,11
- None of these
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Solution:
Let u = a−3d, v =a−d, y = a+d and z = a+3d. Now, sum of producers in first equation,
(a − 3b) + (a − d) = 2
2a − 4 d = 2
sum of products in second equation,
(a + d) + (a + 3d) = 18
2a+4d=18
solving Equation (i) and (ii)
a=5,d=2
u= 5−2×3=−1
v=5−2=3
y=5+2=7
z = 5 + 2 × 3 = 11
Ques 4: If the roots of the equation (a2 + b2 ) x + 2(b2 + c2 )x + (b2 + c2 ) = 0 are real, which of the following must hold true? (CAT 2012)
- c2 ≥ a2
- c4 ≥ a2(b2+c2)
- b2 ≥ a2
- a4 ≤ b2(a2+c2)
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Solution: (A)
As the roots of the equation
(a2 + b2)x2 + 2(b2 + c2)x + (b2 + c2)= 0 are real.
[2(b2 + c2)]2 −4(a2 + b2)(b2 + c2)] ≥ 0
(b2 + c2)−(a2 + b2) ≥ 0
c2 ≥ a2
Ques 5: p is a prime and m is a positive integer. How many solutions exist for the equation p6 − p = (m2 + m + 6)(p−1)? (CAT 2012)
- 0
- 1
- 2
- Infinite
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Solution: (B)
\(\frac{p^6 - p}{p - 1}\) = p5 + p4 + p3 + p2 + p
p5 + p4 + p3 + p2 + p = m (m + 1) + 6.
RHS is even as m or m + 1 is even, the first term is even and the second term is even.
only even primes is 2 so p = 2.
Substituting, we get m2 + m − 56 = 0
(m+8)(m−7)=0
∴ m = − 8 or 7
But m is a positive integer ⇒ m = 7
Therefore, p = 2 and m = 7 is the only solution.
Ques 6: If α and β are the roots of the quadratic equation x2 − 10x + 15 = 0, then find the quadratic equation whose roots are \( (\alpha + \frac{\alpha}{\beta}) \) and \((\beta + \frac{\beta }{\alpha})\) . (CAT 2012)
- 15x2+71x+210=0
- 5x2−22x+56=0
- 3x2 − 44x + 78= 0
- Cannot be determined
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Solution: (C)
The required equation is
x2 – 44/3 x + 26 = 0
3x2 − 44x + 78 = 0
Ques 7: If ax2 + bx + c = 0 and 2a, b and 2c are in arithmetic progression, which of the following are the roots of the equation (CAT 2012)
- a,c
- −a,− c
- −\(\frac{a}{2}\),−\(\frac{c}{2}\)
- −\(\frac{c}{2}\),−1
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Solution:- (D)
2a, b and 2c are in arithmetic progression
2b=2a+2c
b = a+c
Ques 8: The equation,2x2 +2(p+1)x+ p=0, where p is real, always has roots that are (CAT 2010)
- Equal
- Equal in magnitude but opposite in sign
- Irrational
- Real
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Solution: (D)
The value of the discriminant of a quadratic equation will determine the nature of the roots of a quadratic equation.
The discriminant of a quadratic equation ax2 + bx+ c=0 is given by b2 −4ac.
a=2
b=2(p+1)
c=p
D= [2(p + 1)]2 −4x2xp
=4(p + 1)2 −8p
=4[(p+1)2 −2p]
=4[(p2 +2p+1)−2p]
=4(p2 +1)
For any real value of p,4(p2 + 1) will always be
positive as p2 cannot be negative for real p.
discriminant b2 − 4ac will always be positive.
With D>0 , roots will be real.
Ques 9: If the roots of the quadratic equation y2 + My + N are equal to N and M, then find the possible number of pairs of (M, N). (CAT 2009)
- 0
- 1
- 2
- 3
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Solution: (C)
Since, M and N are the roots of y2 + My + N = 0 M + N = − M and MN = N
MN = N ⇒ N = 0 or M = 1
If N = 0 , then M = − M
⇒ M = 0
If M = 1, then N = − 2M
⇒ N = − 2
two (M , N ) pairs, (0, 0) and (1, − 2) are possible.
Ques 10: If the roots of the equation x3 − ax2 + bx − c = 0 are three consecutive integers, then what is the smallest possible value of b ? (CAT 2008)
- - \(\frac{1}{\sqrt 3}\)
- - 1
- 0
- 1
- \(\frac{1}{\sqrt 3}\)
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Solution: (B)
Let the roots of the equation x − ax + bx − c = 0 be
(α −1),α (α +1)
⇒ α(α−1)+α(α+1)+(α+1)(α−1)=b
⇒ α2−α+α2+α+α2−1=b
⇒ 3α2−1=b
Minimum value of b is −1 when α = 0.
Ques 11: Let u = (log2 x)2 − 6 (log2 x) + 12, where x is a real number. Then, the equation xn = 256, has (CAT 2004)
- No solution for x
- Exactly one solution for x
- Exactly two distinct solutions for x
- Exactly three distinct solutions for x
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Solution: (B)
u = (log2x)2−6(log2x)+12
Let log2 x = p ...(i)
⇒ u = p2 − 6 p + 12
xu =256(=28)
Applying log to base 2 on both sides, we get
ulog2 x = log2 28, ulog2 x = 8 ...(ii) Dividing Eq. (ii) by Eq, (i), we get
u = 8 / p
⇒ 8 / p = p2 − 6 p + 12
8−p3 −6p2 +12p or p3 −6p2 +12p−8=0
(p−2)3 =0
p = 2
log2 x = 2 ⇒ x = 22 = 4
So only one solution.
Ques 12: Let p and q be the roots of the quadratic equation x2 − (α − 2)x − α − 1 = 0. What is the minimum possible value of p2 + q2 ? (CAT 2003)
- 5
- 4
- 2
- 3
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Solution: (A)
Sum of roots, p + q = α − 2
Product of roots, pq = − α −1
Now, p2+q2 = (p+q)2−2pq = (α−2)2+2(α+1)
= α 2 + 4 − 4α + 2α + 2 = (α + 1)2 + 5
Hence, the minimum value of this will be 5.
Ques 13: The number of roots common between the two equations x3 +3x2 +4x+5 = 0 and x3 +2x2 +7x+3 = 0 is (CAT 2003)
- 0
- 1
- 2
- 3
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Solution: (A)
x2−3x+2=0
⇒ (x−1)(x−2)=0
If there was a common root between the two, it would be the root of the subtracted equation, hence the roots 1 and 2 do not fulfil any of the original equations.
So, no root.
Ques 14: If 13 x + 1 < 2 z and z + 3 = 5 y2 , then (CAT 2003)
- x is necessarily less than y
- x is necessarily equal to y
- x is necessarily greater than y
- None of the above
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Solution: (D)
13x+1 < 2z
z+3 = 5y2
⇒ 13 x + 1 < 2 (5 y2 − 3 )
⇒ 13x+ 7 < 10y2
⇒10y2 >13x+ 7
In the above equation, all the options (a), (b) and (c) are possible.
Ques 15: If the equation x3 − ax2 + bx − a = 0 has three real roots, then the following is true (CAT 2000)
- a = 11
- a \(\neq\) 1
- b = 1
- b \(\neq\) 1
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Solution: (D)
Since there are three sign changes in the given equation, there can only be a maximum of three positive roots. There is no change in sign if f (-x). Therefore, if,, and, are the roots, then they are all positive and we have, so there is no true negative root.
f(x) = (x − α )(x − β)(x − γ ) = 0
x2 −α+β+γx2 +αβ+βγ+γαx−αβγ
b=αβ+βγ+γα⇒a=α+β+γ=αβγ
=(α +β+ γ)/αβγ =1
= \(\frac{1}{\alpha \beta} + \frac{1}{\alpha \gamma} + \frac{1}{\beta \gamma} = 1\)
αβ, αγ, βγ > 1
⇒ b > 3
How to approach questions on Quadratic equation
- Go through the fundamentals of quadratic equation
- It is quite important to know the nature of roots before diving into the questions.
- Basic Understanding of maxima and minima is essential
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