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Determine the nature of the roots of the following quadratic equations.
(i) 2x^{2} – 3x + 5 = 0
(ii) 2x^{2} - 6x + 3 = 0
(iii) For what value of k (4 - k)x^{2 }+ (2k + 4)x + (8k + 1) = 0 is a perfect square.
(iv) Find the least positive value of k for which the equation x^{2 }+ kx + 4 = 0 has real roots.
(v) Find the value of k for which the given quadratic equation has real roots and distinct roots.
Kx^{2 }+ 2x + 1 = 0
(vi) Kx^{2} + 6x + 1 = 0
(vii) x^{2 }- kx + 9 = 0
The given quadratic equation is in the form of ax^{2 }+ bx + c = 0
So a = 2, b = – 3, c = 5
We know, determinant (D) = b^{2} - 4ac = (-3)^{2} - 4(2)(5) = 9 - 40 = – 31 < 0
Since D < 0, the determinant of the equation is negative, so the expression does not having any real roots.
(ii) 2x^{2} - 6x + 3 = 0 The given quadratic equation is in the form of ax^{2 }+ bx + c = 0
So a = 2, b = -6, c = 3
We know, determinant (D) = b^{2} - 4ac = (- 6)^{2} - 4(2)(3) = 36 - 24 = 12 < 0
Since D > 0, the determinant of the equation is positive, so the expression does having any real and distinct roots
The given equation is (4 - k)x^{2 }+ (2k + 4)x + (8k + 1) = 0
Here, a = 4 - k, b = 2k + 4, c = 8k + 1
The discriminate (D) = b^{2 }- 4ac = (2k+4)^{2} - 4(4 - k)(8k + 1)
= (4k^{2 }+ 16 + 16k) - 4(32k + 4 - 8k^{2 }- k)
= 4(k^{2} + 8k^{2 }+ 4k - 31k + 4-4)
= 4(9k^{2 }- 27k)
D = 4(9k^{2 }- 27k)
The given equation is a perfect square D = 0
4(9k^{2 }- 27k) = 0
9k^{2 }- 27k = 0
Taking out common of 3 from both sides and cross multiplying = k^{2 }- 3k = 0
= K (k - 3) = 0
Either k = 0 Or k = 3
The value of k is to be 0 or 3 in order to be a perfect square.
(iv)Find the least positive value of k for which the equation x^{2 }+ kx + 4 = 0 has real roots.
The given equation is x^{2 }+ kx + 4 = 0 has real roots Here, a = 1, b = k, c = 4
The discriminate (D) = b^{2 }- 4ac = 0 = k^{2} - 16 = 0 = k = 4, k = – 4
The least positive value of k = 4 for the given equation to have real roots.
(v) Find the value of k for which the given quadratic equation has real roots and distinct roots. Kx^{2 }+ 2x + 1 = 0
The given equation is Kx^{2 }+ 2x + 1 = 0
Here, a = k, b = 2, c = 1
The discriminate (D) = b^{2 }- 4ac = 0
= 4 - 4k = 0
= 4k = 4
K = 1
The value of k = 1 for which the quadratic equation is having real and equal roots. (vi) Kx^{2} + 6x + 1 = 0
The given equation is Kx^{2 }+ 6x + 1 = 0
Here, a = k, b = 6, c = 1
= 36 - 4k = 0
= 4k = 36
K = 9
The value of k = 9 for which the quadratic equation is having real and equal roots.
The given equation is x^{2 }- kx + 9 = 0
Here, a = 1, b = – k, c = 9
Given that the equation is having real and distinct roots.
Hence, the discriminate (D) = b^{2 }- 4ac = 0
= k^{2} - 4(1)(9) = 0
= k^{2 }– 36 = 0
= K = – 6 and k = 6
The value of k lies between -6 and 6 respectively to have the real and distinct roots.
Find the value of k (i) Kx^{2 }+ 4x + 1 = 0.
(ii)
(iii) 3x^{2 }- 5x + 2k = 0
(iv) 4x^{2 }+ kx + 9 = 0
(v) 2kx^{2 }- 40x + 25 = 0
(vi) 9x^{2 }- 24x + k = 0
(vii) 4x^{2}- 3kx + 1 = 0
(viii) x^{2 }- 2(5 + 2k)x + 3(7 + 10k) = 0
(ix) (3k +1)x^{2}+ 2(k +1)x + k = 0
(x) Kx^{2 }+ kx + 1 = – 4x^{2 }- x
(xi) (k + 1)x^{2 }+ 2(k + 3)x + k + 8 = 0
(xii) x^{2 }- 2kx + 7k - 12 = 0
(xiii) (k + 1)x^{2 }- 2(3k + 1)x + 8k + 1 = 0
(xiv) 5x^{2 }- 4x + 2 + k(4x^{2 }- 2x + 1) = 0
(xv) (4 - k)x^{2 }+ (2k + 4)x + (8k + 1) = 0
(xvi) (2k + 1)x^{2 }+ 2(k + 3)x + (k +5 ) = 0
(xvii) 4x^{2 }- 2(k + 1)x + (k + 4) = 0
The given equation Kx^{2 }+ 4x + 1 = 0 is in the form of ax^{2 }+ bx + c = 0
Where a = k, b = 4, c = 1
Given that, the equation has real and equal roots D = b^{2 }- 4ac = 0
= 4^{2 }- 4(k)(1) = 0
= 16 – 4k = 0
= k = 4
The value of k is 4
The given equationis in the form of ax^{2 }+ bx + c = 0 where a= k, b = - 2√5, c = 4.
Given that, the equation has real and equal roots D = b^{2}- 4ac = 0
20 - 16k = 0
K = 5 /4
The value of k is k = 5/4
The given equation 3x^{2 }- 5x + 2k = 0 is in the form of ax^{2 }+ bx + c = 0 where a = 3, b = – 5, c = 2k
= (- 5)^{2 }- 4(3)(2k) = 0
= 25 - 24k = 0
K = 25/24
The value of the k is k = 25/24
The given equation 4x^{2 }+ kx + 9 = 0 is in the form of ax^{2 }+ bx + c = 0 where a = 4, b = k, c = 9
= k^{2 }- 4(4)(9) = 0
= k^{2 }- 144 = 0
= k = 12
The value of k is 12
The given equation 2kx^{2 }- 40x + 25 = 0 is in the form of ax^{2 }+ bx + c = 0 where a = 2k, b = – 40, c = 25
(-40)^{2 }- 4(2k)(25) = 0
1600 - 200k = 0
k = 8
The value of k is 8
The given equation 9x^{2 }- 24x + k = 0 is in the form of ax^{2 }+ bx + c = 0 where a = 9, b = – 24, c = k
( – 24)^{2 }- 4(9)(k) = 0
576 - 36k = 0
k = 16
The value of k is 16
The given equation 4x^{2 }- 3kx + 1 = 0 is in the form of ax^{2 }+ bx + c = 0 where a = 4, b = - 3k, c = 1
= (-3k)^{2 }- 4(4)(1) = 0
= 9k^{2 }- 16 = 0
K = 4/3
The value of k is 4/3
The given equation X^{2 }- 2(5 + 2k)x + 3(7 + 10k) = 0 is in the form of ax^{2 }+ bx + c = 0 where a = 1, b = +2(52k), c = 3(7 + 10k)
Given that, the nature of the roots of the equation are real and equal roots D = b^{2 }- 4ac = 0
= (+2(52k))^{2 }- 4(1)(3(7 + 10k)) = 0
= 4(5 + 2k)^{2 }- 12(7 + 10k) = 0
= 25 + 4k^{2 }+ 20k - 21 - 30k = 0
= 4k^{2 }- 10k + 4 = 0
Simplifying the above equation.
We get, = 2k^{2 }- 5k + 2 = 0
= 2k^{2 }- 4k - k + 2 = 0
= 2k(k - 2) - 1(k - 2) = 0
= (k - 2)(2k - 1) = 0, K = 2 and k = 1/2 The value of k can either be 2 or 1/2 (ix) (3k +1)x^{2}+ 2(k +1)x + k = 0
The given equation (3k + 1)x^{2 }+ 2(k + 1)x + k = 0 is in the form of ax^{2 }+ bx + c = 0 where a = 3k + 1, b = +2(k + 1), c = (k)
= [2(k + 1)]^{2 }- 4(3k + 1)(k) = 0
= (k + 1)^{2 }- k(3k + 1) = 0
= -2k^{2 }+ k + 1 = 0
This equation can also be written as 2k^{2 }- k - 1 = 0
The value of k can be obtained by k
The value of k are 1 and (-1)/2 respectively. (x) Kx^{2 }+ kx + 1 = -4x^{2 }- x
Bringing all the x components on one side we get, x^{2}(4 + k) + x(k + 1) + 1 = 0
The given equation Kx^{2 }+ kx + 1 = -4x^{2 }- x is in the form of ax^{2 }+ bx + c = 0 where a = 4 + k,b = +k + 1, c = 1 Given that, the nature of the roots of the equation are real and equal roots D= b^{2}- 4ac = 0
= (k+1)^{2}- 4(4 + k)(1) = 0
= k^{2}- 2k – 10 = 0
The equation is also in the form ax^{2 }+ bx + c = 0
The value of k is obtained by a = 1, b = -2, c = – 15
Putting the respective values in the above formula we will obtain the value of k
The value of k are 5 and -3 for different given quadratic equation.
The given equation (k + 1) x^{2 }+ 2(k + 3)x + k + 8 = 0 is in the form of ax^{2 }+ bx + c = 0 where a = k + 1,b = 2(k + 3), c = k + 8
Given the nature of the roots of the equation are real and equal. D = b^{2 }- 4ac = 0
= [2(k + 30]^{2 }- 4(k + 1)(k + 8) = 0
= 4(k + 3)^{2 }- 4(k + 1)(k + 8) = 0
Taking out 4 as common from the LHS of the equation and dividing the same on the RHS = (k + 3)^{2}-(k + 1)(k + 8) = 0
= k^{2 }+ 9 + 6k - (k^{2 }+ 9k + 18) = 0
Cancelling out the like terms on the LHS side = 9 + 6k - 9k - 8 = 0
= - 3k + 1 = 0
= 3k = 1
K = 1/3
The value of k of the given equation is k =1/3
The given equation is x^{2 }- 2kx + 7k - 12 = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 where a = 1, b = – 2k, c = 7k – 12
= (2k)^{2 }- 4(1)(7k - 12) = 0
= 4k^{2 }- 28k + 48 = 0
= k^{2 }- 7k + 12 = 0
The value of k can be obtained by
Here a = 1, b = – 7k, c = 12
By calculating the value of k is
The value of k for the given equation is 4 and 3 respectively.
The given equation is (k + 1)x^{2 }- 2(3k + 1)x + 8k + 1 = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 where a = k + 1, b = – 2(k + 1), c = 8k + 1
Given the nature of the roots of the equation are real and equal. D = b^{2}- 4ac = 0
= (-2(k + 1))^{2 }- 4(k + 1)(8k + 1) = 0
= 4(3k + 1)^{2 }- 4(k + 1)(8k + 1) = 0
Taking out 4 as common from the LHS of the equation and dividing the same on the
RHS = (3k + 1)^{2 }- (k + 1)(8k + 1) = 0
= 9k^{2 }+ 6k + 1 – (8k^{2 }+ 9k + 1) = 0
= 9k^{2 }+ 6k + 1 – 8k^{2 }- 9k - 1 = 0
= k^{2 }- 3k = 0
= k(k - 3) = 0
Either k = 0 Or, k - 3 = 0 = k = 3
The value of k for the given equation is 0 and 3 respectively.
The given equation 5x^{2 }- 4x + 2 + k(4x^{2 }- 2x + 1) = 0 can be written as x^{2}(5 + 4k) - x(4 + 2k) + 2 - k = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 where a = 5 + 4k, b = -(4 + 2k), c = 2 - k
Given the nature of the roots of the equation are real and equal.
D = b^{2 }- 4ac = 0
= [-(4 + 2k)]^{2 }- 4(5 + 4k)(2 - k) = 0
= 16 + 4k^{2 }+ 16 - 4(10 - 5k + 8k - 4k^{2}] = 0
= 16 + 4k^{2 }+ 16 - 40 + 20k - 32k + 16k^{2} = 0
= 20k^{2 }- 4k - 24 = 0
Taking out 4 as common from the LHS of the equation and dividing the same on the RHS = 5k^{2 }- k - 6 = 0 The value of k can be obtained by equation
The value of k for the given equation are k = 6/5 and−1 respectively.
The given equation is in the form of ax^{2 }+ bx + c = 0 where a = 4 - k, b = (2k + 4), c = 8k + 1
= (2k + 4)^{2 }- 4(4 - k)(8k + 1) = 0
= 4k^{2 }+ 16k + 16 - 4(-8k^{2 }+ 32k + 4 - k) = 0
= 4k^{2 }+ 16k + 16 + 32k^{2 }- 124k - 16 = 0
Cancelling out the like and opposite terms.
We get, = 36k^{2 }- 108k = 0
Taking out 4 as common from the LHS of the equation and dividing the same on the RHS = 9k^{2 }- 27k = 0
= 9k(k -3 ) = 0
Either 9k = 0 K = 0 Or, k - 3 = 0 K = 3
(xvi) (2k + 1)x^{2 }+ 2(k + 3)x + (k +5 )= 0
The given equation is (2k + 1)x^{2 }+ 2(k + 3)x + (k + 5) = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 where a = 2k + 1, b = 2(k + 3), c = k + 5
= [2(k + 3)]^{2 }- 4(2k + 1)(k + 5) = 0
RHS = [(k + 3)]^{2 }- (2k + 1)(k + 5) = 0
= K^{2 }+ 9 + 6k - (2k^{2 }+ 11k + 5) = 0
= – k^{2 }- 5k + 4 = 0
= k^{2 }+ 5k - 4 = 0
The value of k can be obtained by k = 6/5 and − 1 respectively.
Here a = 1, b = 5, c = – 4
The value of k for the given equation is
The given equation is 4x^{2 }- 2(k + 1)x + (k + 4) = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 where a = 4, b = -2(k + 1), c = k + 4
= [-2(k + 1)]^{2 }- 4(4)(k + 4) = 0
RHS = (k + 1)^{2 }- 4(k + 4) = 0
= k^{2 }+ 1 + 2k - 4k - 16 = 0
= k^{2 }- 2k - 15 = 0
Here a = 1, b = -2, c = -15
In the following, determine the set of values of k for which the given quadratic equation has real roots:
(i) 2x^{2 }+ 3x + k = 0
(ii) 2x^{2 }+ kx + 3 = 0
(iii) 2x^{2 }- 5x - k = 0
(iv) Kx^{2 }+ 6x + 1= 0
(v) x^{2 }- kx + 9 = 0
The given equation is 2x^{2 }+ 3x +k = 0
The given quadratic equation has equal and real roots D = b^{2}- 4ac = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 so, a = 2, b = 3, c = k = 9 - 4(2)(k) = 0
= 9 - 8k = 0
= k ≤ 98
The value of k does not exceed k ≤ 98 to have a real root.
The given equation is 2x^{2 }+ kx + 3 = 0
The given quadratic equation has equal and real roots D = b^{2 }- 4ac = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 so, a = 2, b = k, c = 3 = k^{2 }- 4(2)(3) = 0
= k^{2 }– 24 = 0
The value of k should not exceedin order to obtain real roots.
The given equation is 2x^{2 }- 5x - k = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 so, a = 2, b = - 5, c = - k = 25 - 4(2)(- k) = 0
= 25 - 8k = 0
= k ≤ 25/8
The value of k should not exceed k ≤ 25/8 (iv) Kx^{2 }+ 6x + 1= 0
The given equation is in the form of ax^{2 }+ bx + c = 0 so, a = k, b = 6, c = 1 = 36 - 4(k)(1) = 0
= k = 9
The value of k for the given equation is k = 9
The given equation is X^{2 }- kx + 9 = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 so, a = 1, b = -k, c = 9 = k^{2 }- 4(1)(-9) = 0
= k^{2 }= 36 k ≥ √36 K = 6 and k = - 6
The value of k should in between K = 6 and k = - 6 in order to maintain real roots.
(i) 2x^{2} - 3x + 5 = 0
(iii) For what value of k (4 - k)x^{2 }+ (2k + 4)x + (8k + 1) = 0 is a perfect square
(vii) x^{2 }– kx + 9 = 0
So a = 2, b = -3, c = 5
We know, determinant (D) = b^{2} – 4ac = (-3)^{2} – 4(2)(5)
= 9 – 40 = – 31 < 0
We know, determinant (D) = b^{2} – 4ac = (-6)^{2} - 4(2)(3) = 36 – 24 = 12 < 0
Since D > 0, the determinant of the equation is positive, so the expression does having any real and distinct roots. (iii) For what value of k (4 - k)x^{2 }+ (2k + 4)x + (8k + 1) = 0 is a perfect square
The discriminate (D) = b^{2 – }4ac
= (2k + 4)^{2} – 4(4 - k)(8k + 1)
= 4(k^{2 }+ 8k^{2 }+ 4k - 31k + 4 - 4)
Taking out common of 3 from both sides and cross multiplying K^{2 }- 3k = 0 K (k - 3) = 0
The discriminate (D) = b^{2 – }4ac = 0
= k^{2} – 16 = 0
= k = 4, k = - 4
The value of k = 1 for which the quadratic equation is having real and equal roots.
= K = 9
The given equation is X^{2 }– kx + 9 = 0
Here, a = 1, b = -k, c = 9
Given that the equation is having real and distinct roots. Hence, the discriminate (D) = b^{2 – }4ac = 0
= k^{2} – 4(1)(9) = 0
= k^{2 }- 36 = 0
= k = – 6 and k = 6
Find the values of k for which the given quadratic equation has real and distinct roots.
(i) Kx^{2 }+ 2x + 1 = 0
(ii) Kx^{2 }+ 6x + 1 = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 so, a = k, b = 2, c = 1 D = b^{2 }- 4ac = 0
= 4 - 4(1)(k) = 0
k = 1
The value of k for the given equation is k = 1 (ii) Kx^{2 }+ 6x + 1 = 0
The given equation is in the form of ax^{2 }+ bx + c = 0 so, a = k, b = 6, c = 1
= 36 - 4(1)(k) = 0
For what value of k, (4 - k)x^{2 }+ (2k + 4)x + (8k + 1) = 0, is a perfect square.
The given equation is in the form of ax^{2 }+ bx + c = 0 so, a = 4 - k, b = 2k + 4, c = 8k + 1 D = b^{2 }- 4ac
= (2k + 4)^{2 }- 4(4 - k)(8k + 1)
= 4k^{2 }+ 16 + 4k - 4(32 + 4 - 8k^{2 }- k)
= 4(k^{2 }+ 4 + k - 32 - 4 + 8k^{2 }+ k)
=4(9k^{2 }- 27k)
Since the given equation is a perfect square Therefore D = 0 = 4(9k^{2 }- 27k) = 0
= (9k^{2 }- 27k) = 0
= 3k(k - 3) = 0
Therefore 3k = 0
K = 0 Or, k-3 = 0 K = 3 The value of k should be 0 or 3 to be perfect square.
If the roots of the equation (b - c)x^{2} + (c - a)x + (a - b) = 0 are equal , then prove that 2b = a + c.
The given equation is (b - c)x^{2} + (c - a)x + (a - b) = 0.
The given equation is the form of ax^{2 }+ bx + c = 0.
So, a = (b - c), b = (c - a), c = (a - b)
According to question the equation is having real and equal roots.
Hence discriminant (D) = b^{2 }- 2ac = 0
= (c - a)^{2 }- 4(b - c)(a - b) = 0
= c^{2 }+ a^{2 }- 2ac – 4(ab - b^{2 }- ac + cb) = 0
= c^{2 }+ a^{2 }- 2ac – 4ab + 4b^{2 }+ 4ac - 4cb = 0
= c^{2 }+ a^{2 }+ 2ac – 4ab + 4b^{2 }- 4cb = 0
= (a + c)^{2 }- 4ab + 4b^{2 }- 4cb = 0
= (c + a - 2b)^{2} = 0
= (c + a - 2b) = 0
= c + a = 2b
Hence it is proved that c + a = 2b.
If the roots of the equation (a^{2 }+ b^{2}) x^{2 }– 2(ac + bd)x + (c^{2 }+ d^{2}) = 0 are equal. Prove that a ÷ b = c ÷ d.
The given equation is (a^{2 }+ b^{2})x^{2} – 2(ac + bd)x + (c^{2 }+ d^{2}) = 0.
The equation is in the form of ax^{2 }+ bx = c = 0
Hence, a = (a^{2 }+ b^{2}), b = – 2(ac + bd), c = (c^{2 }+ d^{2}).
The given equation is having real and equal roots.
Discriminant (D) = b^{2 }- 4ac = 0
= [-2(ac + bd)]^{2} - 4 (a^{2 }+b^{2})(c^{2 }+ d^{2}) = 0
= (ac + bd)^{2} - (a^{2 }+ b^{2})(c^{2 }+ d^{2}) = 0
= a^{2}c^{2} + b^{2}d^{2 }+ 2abcd – (a^{2}c^{2} + a^{2}d^{2} + b^{2}c^{2 }+ b^{2}d^{2}) = 0
Cancelling out the equal and opposite terms.
We get, = 2abcd - a^{2}d^{2} - b^{2}c^{2 }= 0
= abcd + abcd - a^{2}d^{2} - b^{2}c^{2 }= 0
= ad(bc - ad) + bc(ad - bc) = 0
= ad(bc - ad) - bc(bc - ad) = 0
= (ad - bc)(bc - ad) = 0
= ad – bc = 0
= (a ÷ b) = (c ÷ d)
Hence, it is proved.
If the roots of the equation ax^{2 }+ 2bx + c = 0 andare simultaneously real, then prove that b^{2 }- ac = 0.
The given equations are ax^{2 }+ 2bx +c = 0 and
These two equations are of the form ax^{2 }+ bx + c = 0.
Given that the roots of the two equations are real.
Hence, D = 0 that is b^{2 }- 4ac = 0
Let us assume that ax^{2 }+ 2bx + c = 0 be equation (i) andbe (ii) From equation (i) b^{2 }- 4ac = 0
= 4 b^{2 }- 4ac = 0 .... (iii)
From equation (ii) b^{2 }- 4ac = 0
Given, that the roots of equation (i) and (ii) are simultaneously real and hence equation (iii) = equation (iv).
= 4b^{2 }- 4ac = 4ac - 4
b^{2} = 8ac = 8b^{2} = b^{2 }- ac = 0.
Hence it is proved that b^{2 }- ac = 0.
If p, q are the real roots and p = q. Then show that the roots of the equation (p-q)x^{2 }+ 5(p + q)x - 2(p - q) = 0 are real and equal.
The given equation is (p - q)x^{2} + 5(p + q)x - 2(p - q) = 0
Given, p , q are real and p ? q.
Then, Discriminant (D) = b^{2}– ac = [5(p + q)]^{2} - 4(p - q)(-2(p - q)) = 25(p + q)^{2} + (p - q)^{2}
We know that the square of any integer is always positive that is, greater than zero.
Hence, (D) = b^{2 }–ac = 0 As given, p, q are real and p = q.
Therefore, = 25(p + q)^{2} + (p - q)^{2 }? 0 = D = 0
Therefore, the roots of this equation are real and unequal.
If the roots of the equation (c^{2 }- ab)x^{2 }- 2(a^{2 }- bc)x + b^{2 }- ac = 0 are equal , then prove that either a = 0 or a³ + b³ + c³ = 3abc .
The given equation is (c^{2 }- ab)x^{2 }- 2(a^{2 }- bc)x + b^{2 }- ac = 0
This equation is in the form of ax^{2 }+ bx + c = 0
So, a = (c^{2 }- ab), b = -2(a^{2 }- bc), c = b^{2 }- ac.
According to the question, the roots of the given question are equal.
Hence, D= 0, b^{2 }- 4ac = 0
= [-2(a^{2 }- bc)]^{2} - 4(c^{2 }- ab)( b^{2 }- ac) = 0
= 4(a^{2 }- bc)^{2} - 4(c^{2 }- ab)( b^{2 }- ac) = 0
= 4a(a³ + b³ + c³ - 3abc) = 0
Either 4a =0 therefore, a = 0 Or, (a³ + b³ + c³ - 3abc) = 0
= (a³ + b³ + c³) = 3abc Hence its is proved.
Show that the equation 2(a^{2 }+ b^{2})x^{2 }+ 2(a + b)x + 1 = 0 has no real roots , when a = b.
The given equation is 2(a^{2 }+ b^{2})x^{2 }+ 2(a + b)x - 1 = 0
Here, a = 2(a^{2 }+ b^{2}), b = 2(a + b), c = +1.
Given, a = b The discriminant (D) = b^{2 }- 4ac = [2(a + b)]^{2} - 4(2(a^{2 }+ b^{2}))(1)
= 4(a + b)^{2} - 8(a^{2 }+ b^{2})
= 4(a^{2 }+ b^{2 }+ 2ab) – 8a^{2 }- 8b^{2}
= +2ab – 4a^{2 }- 4b^{2}
According to the question a = b, as the discriminant D has negative squares so the value of D will be less than zero. Hence, D = 0, when a = b.
Prove that both of the roots of the equation (x - a)(x - b) +(x - c)(x - b) + (x - c)(x - a) = 0 are real but they are equal only when a = b = c.
The given equation is (x - a)(x - b) + (x - c)(x - b) + (x - c)(x - a) = 0
By solving the equation, we get it as, 3x^{2 }- 2x(a + b + c) + (ab + bc + ca) = 0
Here, a = 3, b = 2(a + b + c), c = (ab + bc + ca)
The discriminate (D) = b^{2 }- 4ac
= [-2(a + b + c)]^{2 }- 4(3)(ab + bc + ca)
= 4(a + b +c )^{2 }-12(ab + bc + ca)
= 4[(a + b + c)^{2 }- 3(ab + bc + ca)]
= 4[a^{2 }+ b^{2 }+ c^{2 }– ab - bc - ca]
= 2[2a^{2 }+ 2b^{2 }+ 2c^{2 }– 2ab - 2bc - 2ca]
= 2[(a - b)^{2 }+ (b - c)^{2 }+ (c - a)^{2}]
Here clearly D = 0, if D = 0 then, [(a - b)^{2 }+ (b - c)^{2 }+ (c - a)^{2}] = 0
a –b = 0 b – c = 0 c – a = 0
Hence, a = b = c = 0 Hence, it is proved.
If a, b, c are real numbers such that ac = 0, then, show that at least one of the equations ax^{2 }+ bx + c = 0 and – ax^{2 }+ bx + c = 0 has real roots.
The given equation are ax^{2 }+ bx + c = 0 … (i)
And - ax^{2 }+ bx + c = 0 … (ii)
Given, equations are in the form of ax^{2}+ bx + c = 0 also given that a, b, c are real numbers and ac = 0.
The Discriminant (D) = b^{2 }- 4ac For equation (i) = b^{2 }- 4ac ... (iii)
For equation (ii) = b^{2 }- 4(-a)(c) = b^{2 }+ 4ac … (iv)
As a, b, c are real and given that ac = 0
Hence b^{2 }- 4ac = 0 and b^{2}+ 4ac = 0 Therefore, D = 0 Hence proved.
If the equation (1 + m^{2})x + 2mcx + (c^{2 }- a^{2}) = 0 has real and equal roots , prove that c^{2 }= a^{2}(1 + m^{2}).
The given equation is (1 + m^{2})x^{2 }+ 2mcx + (c^{2 }- a^{2}) = 0
The above equation is in the form of ax^{2 }+ bx + c = 0.
Here a = (1 + m^{2}), b = 2mc, c = +(c^{2 }- a^{2})
Given, that the nature of the roots of this equation is equal and hence D = 0, b^{2 }- 4ac = 0
= (2mc)^{2} - 4(1 + m^{2})(c^{2 }- a^{2}) = 0
= 4m^{2}c^{2 }– 4(c^{2 }+ m^{2}c^{2 }- a^{2} – a^{2}m^{2}) = 0
= 4(m^{2}c^{2 }- c^{2 }+ m^{2}c^{2 }+ a^{2} + a^{2}m^{2}) = 0
= m^{2}c^{2 }- c^{2 }+ m^{2}c^{2 }+ a^{2} + a^{2}m^{2 }= 0
Now cancelling out the equal and opposite terms, = a^{2} + a^{2}m^{2 }- c^{2 }= 0
= a^{2} (1 + m^{2}) – c^{2} = 0
Therefore, c^{2 }= a^{2} (1 + m^{2}) Hence it is proved that as D = 0, then the roots are equal of c^{2 }= a^{2} (1+ m^{2}).
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Chapter 8: Quadratic Equations Exercise –...