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Title: Algebra _ Roots of the quadratic equation
Description: Algebra _ Roots of the quadratic equation Roots of Quadratic Equation. The values of variables satisfying the given quadratic equation are called its roots. In other words, x = α is a root of the quadratic equation f(x), if f(α) = 0. The real roots of an equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersect the x-axis.
Description: Algebra _ Roots of the quadratic equation Roots of Quadratic Equation. The values of variables satisfying the given quadratic equation are called its roots. In other words, x = α is a root of the quadratic equation f(x), if f(α) = 0. The real roots of an equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersect the x-axis.
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Algebra
Roots of the quadratic equation
I
...
(i) (x 1) (2x – 5) = 0
Solution:
2x2 – 7x + 5 = 0 a = 2, b = 7, c = 5
= b2 – 4ac = (7)2 – 4 (2) (5)
49 – 40
= 9, a perfect square
The roots are real, unequal and rational
...
(iii)
3 x2 – 2 6 x + 2 = 0
Solution:
a = 3, b = 26,
= b2 – 4ac
c = 2
= (26 )2 – 4 (3) (2)
= 24 – 24 = 0
The roots are real and equal
...
(v) (x – 2a) (x – 2b ) = 4ab
Solution:
(x – 2a) (x – 2b)
= 4ab
x2 – (2a + 2b)x + 4ab – 4ab
= 0
x2 – (2a + 2b) x
= 0
a = 1 , b = – (2a + 2b), c
= 0
= b2 – 4ac = – (2a + 2b)2 a prefect square
The roots are real, distinct and rational
(vi) 9 a2 b2 x2 – 24 a b c dx + 16 c2d2 = 0 , a 0 b 0
...
II
...
Find the values of k which the equation kx – 6x – 2 = 0 has real roots
...
If the equation ( 1 + m2) x2 + 2m c x + (c2 – a2) = 0 has equal roots, prove that
c2 = a2 (1 + m2)
Solution:
= 0 for equal roots
4 m2 c2 – 4(1 + m2) (c2 – a2)
= 0
m2 c2 – (1 + m2) (c2 – a2)
= 0
m2 c2 – c2 m2 c2 + a2 + a2 m2
= 0
c2
= a2 + a2 m2
= a2 (1 + m2)
2
...
Solution:
(b – c) x2 + (c – a)x + (a – b)
= 0
For equal roots
= 0
(c – a)2 – 4(b – c) (a – b)
= 0
c2 + a2 2ac – 4 (ab – ac – b2 + bc)
= 0
c2 + a2 2ac – 4ab + 4ac + 4b2 – 4bc
= 0
c2 + a2 + 2ac
= 4ab – 4b2 + 4bc
(c + a)2
= 4 (ab + bc – b2)
(c + a)2
= 4b (a + c) – 4b2
Put a + c
= y
y2 – 4by + 4b2
= 0
(y – 2b)2
Hence a + c
3
...
=
c
d
If the roots of (a b) x2 + (b – c)x + (c a) = 0 are equal, prove that 2a = b + c
Title: Algebra _ Roots of the quadratic equation
Description: Algebra _ Roots of the quadratic equation Roots of Quadratic Equation. The values of variables satisfying the given quadratic equation are called its roots. In other words, x = α is a root of the quadratic equation f(x), if f(α) = 0. The real roots of an equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersect the x-axis.
Description: Algebra _ Roots of the quadratic equation Roots of Quadratic Equation. The values of variables satisfying the given quadratic equation are called its roots. In other words, x = α is a root of the quadratic equation f(x), if f(α) = 0. The real roots of an equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersect the x-axis.