StudentAidUnit 2: Quadratic Equations — Exercise 2 4
10th Class Mathematics · Unit 2: Quadratic Equations
2.4.1.(i) Examine the nature of roots of 3x² - 9x - 2 = 0. (ii) Examine the nature of roots of x² + 6x + 9 = 0. (iii) Examine the nature of roots of 2x² + 4x + 5 = 0. (iv) Examine the nature of roots of 7x² - 6x - 1 = 0. (v) Examine the nature of roots of 5x² - 2x + 10 = 0. (vi) Examine the nature of roots of x² - 8x + 16 = 0.
Given
(i)
a=3, b=-9, c=-2
Working
(i) Discriminant
D=(-9)2-4(3)(-2)=81+24=105>0
Result
(i) Nature
Two real and unequal roots
Given
(ii)
a=1, b=6, c=9
Working
(ii) Discriminant
D=62-4(1)(9)=36-36=0
Result
(ii) Nature
Two real and equal roots
Given
(iii)
a=2, b=4, c=5
Working
(iii) Discriminant
D=42-4(2)(5)=16-40=-24<0
Result
(iii) Nature
Two non-real (imaginary) roots
Given
(iv)
a=7, b=-6, c=-1
Working
(iv) Discriminant
D=(-6)2-4(7)(-1)=36+28=64>0
Result
(iv) Nature
Two real and unequal roots
Given
(v)
a=5, b=-2, c=10
Working
(v) Discriminant
D=(-2)2-4(5)(10)=4-200=-196<0
Result
(v) Nature
Two non-real (imaginary) roots
Given
(vi)
a=1, b=-8, c=16
Working
(vi) Discriminant
D=(-8)2-4(1)(16)=64-64=0
Result
(vi) Nature
Two real and equal roots
2.4.2.For what values of t, the roots of 3x² + x + 9t = 0 are real and unequal?
Given
For real, unequal roots
D>0, a=3, b=1, c=9t
Working
12-4(3)(9t)>0 ⇒ 1-108t>0
Result
t<1108
2.4.3.If the quadratic equation 16x² + 7px + 49 = 0 has equal roots, then find the values of p.
Given
For equal roots
D=0, a=16, b=7p, c=49
Working
(7p)2-4(16)(49)=0 ⇒ 49p2=3136
p2=64
Result
p= ± 8
2.4.4.If the quadratic equation 4u² + 8u + q = 0 has unequal and real roots, find the possible values for q.
Given
For unequal, real roots
D>0, a=4, b=8, c=q
Working
82-4(4)(q)>0 ⇒ 64-16q>0
Result
q<4
2.4.5.Find the value for m, if the quadratic equation mx² - 8x + 1 = 0 has real and equal roots.
Given
For real, equal roots
D=0, a=m, b=-8, c=1
Working
(-8)2-4(m)(1)=0 ⇒ 64-4m=0
Result
m=16