Nature of Roots of a Quadratic Equation

First of all, let’s recall the quadratic formula. Yes, in fact, the nature of roots can be determined by simply the value of an expression. Do you remember what nature of roots of a quadratic equation is? Does it refer to whether the roots are real or not real, equal or unequal?

Discriminant of a Quadratic Equation Consider a quadratic equation: ax2 + bx + c = 0, where a  0 The value of this expression can determine the nature of roots. a 2 ac 4 b x - ± = Its roots are given by:  Quadratic formula The expression is called the discriminant (denoted by ).  pronounced as ‘delta’ Discriminant:  = b2  4ac

ac b 4 - Case 1:  > 0  i.e. b2 – 4ac > 0 ∵ is a positive real number. ∵ ∴ The roots of the quadratic equation are and . a ac b 2 4 - +  The two roots are real and unequal. two unequal real roots

ac b 4 - Case 1:  > 0  i.e. b2 – 4ac > 0 ∵ is a positive real number. ∵ ∴ The roots of the quadratic equation are and . a ac b 2 4 - +  ∴ The equation has two unequal real roots. e.g. Consider x2 + 3x – 7 = 0. = 32 – 4(1)(–7) = 37 > 0 ∴ The equation x2 + 3x – 7 = 0 has two unequal real roots.

ac b 4 - Case 2:  = 0  i.e. b2 – 4ac = 0 ∵ is zero. ∴ The roots of the quadratic equation are . a b 2 - . a b 2  - ac b 4 2 - The two roots are real and equal. one double real root

ac b 4 - Case 2:  = 0  i.e. b2 – 4ac = 0 ∵ is zero. ∴ The roots of the quadratic equation are . a b 2 - ∴ The equation has one double real root. e.g. Consider x2 – 8x + 16 = 0. = (–8)2 – 4(1)(16) = 0 ∴ The equation x2 – 8x + 16 = 0 has one double real root.

ac b 4 - Case 3:  < 0  i.e. b2 – 4ac < 0 is not a real number. ∵ ∴ The roots of the quadratic equation are not real. ∴ The equation has no real roots. The two roots are no real roots.

ac b 4 - Case 3:  < 0  i.e. b2 – 4ac < 0 is not a real number. ∵ ∴ The roots of the quadratic equation are not real. ∴ The equation has no real roots. e.g. Consider x2 – 2x + 5 = 0. = (–2)2 – 4(1)(5) = –16 < 0 ∴ The equation x2 – 2x + 5 = 0 has no real roots.

The table below summarizes the three cases of the nature of roots of a quadratic equation. Condition Nature of its roots  > 0  = 0  < 0 two unequal real roots one double real root no real roots (or two distinct real roots) (or two equal real roots)

Follow-up question Find the value of the discriminant of the equation x2 – 5x + 3 = 0, and hence determine the nature of its roots. The discriminant of x2 – 5x + 3 = 0 is given by: 13 ) 3 )( 1 ( 4 5 2 = - D > ∴ The equation has two distinct real roots.

Graph of a Quadratic Equation For the quadratic equation ax2 + bx + c = 0, are there any relations among the following: discriminant, nature of roots, no. of x-intercepts of the corresponding graph.

For the quadratic equation ax2 + bx + c = 0: 2 unequal real roots 1 double real root 0 real roots Value of the discriminant determine nature of roots of ax2 + bx + c = 0 no. of x-intercepts of the graph of y = ax2 + bx + c 2 x-intercepts 1 x-intercept 0 x-intercepts Roots of ax2 + bx + c = 0 equals no. of x-intercepts of the graph of y = ax2 + bx + c value of the discriminant Therefore also tells us

No. of x-intercepts of the graph of y = ax 2 + bx + c The discriminant of the quadratic equation ax2 + bx + c = 0, the nature of its roots and the number of x-intercepts of the graph of y = ax2 + bx + c have the following relations. Discriminant ( = b 2  4ac) Nature of roots of ax 2 + bx + c = 0 No. of x-intercepts of the graph of y = ax 2 + bx + c 2 unequal real roots 1 double real root no real roots 2 1  > 0  = 0  < 0

Follow-up question Some graphs of y = ax2 + bx + c will be shown one by one. For each corresponding quadratic equation ax2 + bx + c = 0, determine whether  > 0,  = 0 or  < 0. O y x O y x  > 0 2 x-intercepts  > 0 2 x-intercepts O y x O x y 1 x-intercept  < 0  = 0 no x-intercepts