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Quadratic Equations

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Quadratic equations are very familiar to students, since they have already studied them at school level. This topic involves some very basic formulae and concepts that can be used to solve problems.

An equation of the form ax2 + bx + c = 0 where a ¹ 0 is said to be a quadratic equation in the

variable x. The name ‘quadratic’ arises since the highest power of x is 2. Here, a, b and c are real numbers. A typical quadratic equation would be x2 - 5x + 6 = 0 .

ROOTS OF A QUADRATIC EQUATION

A quadratic equation in x will have two values of x that satisfy the equation. These are the roots of the equation. The roots may be real or imaginary.

The roots of a quadratic equation may be found by either of the two methods – by factorization or by applying the formula for roots.

Finding the roots by factorization

The quadratic equation ax+ bx + c = 0 can be written in the form (x - a)(x -b), where α and β are the two roots, by factorizing the given equation. In order to do that, express b (coefficient of x) as the sum of two numbers whose product is equal to ac (product of constant term and coefficient of x2). This can be clearly understood from the example below.

Consider the equation x2 - 5x + 6 = 0 .

This can be rewritten as     x2-(2+3)x+(2×3)=0

i.e.,              x2-2x-3x+6=0

or ,               x(x-2)-3(x-2)=0

(x-3)(x-2)=0

so ,the roots are x = 3 and x = 2.

Consider another equation 2x2+3x-2=0.

This can be rewritten as     2x2+(4-1)x -2 = 0

i.e.,              2x2+4x-x-2=0

or,               2x(x+2)-1(x+2)=0

i.e.,              (2x-1)(x+2) = 0

so, the roots are x = 1/2 and x = -2.

 

Finding the roots by formula

The roots of the quadratic equation ax2 + bx + c = 0 are directly given by the formula

 

The roots found in the two examples earlier may be verified using the formula.

Note: While the formula is almost always a “fail safe” option guaranteed to find the roots of the quadratic equation, it is often time-consuming. Hence, the formula approach must be used only as a last resort. One good situation to use the formula would be when the coefficients are irrational numbers like √2 or √3.

Sum and product of the roots

Sum of the roots, α+β = -b/a

Product of the roots, αβ = c/a

Nature of the roots

The expression b2 -4ac determines the nature of the roots of the quadratic equation and is called the discriminant. Based on the value of b2 -4ac , the following can be said about the roots.

 

b2 -4ac > 0

Roots are real and distinct

b2 -4ac = 0

Roots are real and equal

b2 -4ac < 0

Roots are imaginary and complex conju- gates

 

When a, b and c are rational, the table above is modified if b2 -4ac > 0. The following table is then applicable.

 

b2 -4ac >0 and is a perfect square

Roots are rational and distinct

b2 -4ac > 0 and is not a perfect square

Roots are irrational and distinct

b2 -4ac = 0

Roots are rational and equal

b2 -4ac < 0

Roots are imaginary and will be com- plex conjugates

 

When a, b and c are rational and the roots are irrational, the roots will be conjugate surds, i.e., of the form   . This means that irrational roots occur in conjugate pairs. Thus  if      is a root, then  is also a root

Signs of the roots

Based on the signs of the sum and product of the roots, the following can be observed regarding the signs of the roots.

Sign of product of roots

Sign of sum of roots

Sign of the roots

+

+

Both roots are positive

+

-

Both roots are negative

-

+

Numerically larger root is positive and other is negative

-

-

Numerically larger root is negative and other is positive

 

CONSTRUCTING A QUADRATIC EQUATION 

When the roots of the equation are α and β, the equation is (x -  α)(x -β) = 0 or x2 -(α + β)x + αβ= 0.

When the sum of the roots S and product of the roots P are given, the equation is x2 - Sx +P = 0.

Constructing the equation when relation between the roots of the equation to be framed and another equation is given:

  • Equation whose roots are 1/α and 1/β (i.e., reciprocals of the original roots) is obtained by substituting 1/x in the place of x in the given equation. The new equation is cx2 + bx + a = 0 .
  • Equation whose roots are α+k and β+k (i.e., k more than the original roots) is obtained by substituting x-k in the place of x in the given
  • Equation whose roots are α-k and β-k (i.e., k less than the original roots) is obtained by substituting x+k in the place of x in the given
  • Equation whose roots are kα and kβ (i.e., k times the original roots) is obtained by substituting x/k in the place of x in the given
  • Equation whose roots are α/k and β/k (i.e., 1/k times the original roots) is obtained by substituting kx in the place of x in the given

 

QUADRATIC EXPRESSIONS

A quadratic expression refers to the LHS of a quadratic equation, and is an expression of the form ax2 + bx + c .

 

Maximum and minimum value of a quadratic expression

As x varies from -∞ to ∞ (i.e., when x is real), the quadratic expression ax2 + bx + c

  • Has a minimum value when a >0. This minimum value occurs at x = -b/2a and is equal to     
  • Has a maximum value when a <0. This maximum value occurs at x = -b/2a and is equal to 

It can be observed that the expressions for the value of x and the value of the minimum/ maximum are identical in both cases. It is to be noted that the sign of ‘a’ will determine whether the value is a minimum or a maximum.

Sign of a quadratic expression

Given the sign of a quadratic expression

ax2 + bx + c (> 0 or < 0), the range of values of x satisfying this inequality can be determined.

Assume that α and β are the roots and α < β.

  • When a and ax2+ bx + c are of opposite sign, e., a > 0 and ax2 + bx + c < 0, or a < 0 and ax2 + bx + c > 0, then x lies between the values of α and β, i.e., α < x < β.
  • When a and ax2 + bx + c are of same sign, i.e., a > 0 and ax2 + bx + c > 0, or a < 0 and ax2 + bx + c < 0, then x lies beyond the values of α and β, i.e., x < α or x > β.

EQUATION OF A HIGHER DEGREE

The degree of an equation is the highest power of x present in the equation. If the term having the highest power of x in an equation is x5, then the degree of the equation is 5.

An nth degree equation will have n roots, some of which may be equal. If a is a root of f(x) = 0, i.e., if f(a) = 0, then (x – a) is a factor of f(x).

Types of roots

  • If all the coefficients present in f(x) are real, then complex roots (if any) occur as conjugate pairs. Thus, if p+iq is a root, then p-iq will also be a Thus an equation of odd degree and real coefficients will have at least one real root.
  • If the number of sign changes in f(x) is p, then f(x) = 0 has at most p positive

Eg. f(x) = x3 – 3x2 – 5x + 2. Signs are + - - +. Here, number of sign changes is 2. Hence the equation will have at most 2 positive roots.

  • If the number of sign changes in f(-x) is q, then f(x) = 0 has at most q negative Eg. F (x) = x3 – 3x2 – 5x + 2.

F (-x) = -x3 – 3x2 + 5x + 2.

Signs are - - + +. Here, number of sign changes is 1. Hence, the equation will have at most 1 negative root.


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