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We will solve different types of problems on quadratic equation using quadratic formula and by method of completing the squares. We know the general form of the quadratic equation i.e., ax2 + bx + c = 0, that will help us to find the nature of the roots and formation of the quadratic equation whose roots are given.
1. Solve the quadratic equation 3x2 + 6x + 2 = 0 using quadratic formula.
Solution:
The given quadratic equation is 3x2 + 6x + 2 = 0.
Now comparing the given quadratic equation with the general form of the quadratic equation ax2 + bx + c = 0 we get,
a = 3, b = 6 and c = 2
Therefore, x = βbΒ±βb2β4ac2a
β x = β6Β±β62β4(3)(2)2(3)
β x = β6Β±β36β246
β x = β6Β±β126
β x = β6Β±2β36
β x = β3Β±β33
Hence, the given quadratic equation has two and only two roots.
The roots are β3ββ33 and β3ββ33.
2. Solve the equation 2x2 - 5x + 2 = 0 by the method of completing the squares.
Solutions:
The given quadratic equation is 2x2 - 5x + 2 = 0
Now dividing both sides by 2 we get,
x2 - 52x + 1 = 0
β x2 - 52x = -1
Now adding (12Γβ52) = 2516 on both the sides, we get
β x2 - 52x + 2516 = -1 + 2516
β (xβ54)2 = 916
β (xβ54)2 = (34)2
β x - 54 = Β± 34
β x = 54 Β± 34
β x = 54 - 34 and 54 + 34
β x = 24 and 84
β x = 12 and 2
Therefore, the roots of the given equation are 12 and 2.
3. Discuss the nature of the roots of the quadratic equation 4x2 - 4β3 + 3 = 0.
Solution:
The given quadratic equation is 4x2 - 4β3 + 3 = 0
Here the coefficients are real.
The discriminant D = b2 - 4ac = (-4β3 )2 - 4 β 4 β 3 = 48 - 48 = 0
Hence the roots of the given equation are real and equal.
4. The coefficient of x in the equation x2 + px + q = 0 was taken as 17 in place of 13 and thus its roots were found to be -2 and -15. Find the roots of the original equation.
Solution:
According to the problem -2 and -15 are the roots of the equation x2 + 17x + q = 0.
Therefore, the product of the roots = (-2)(-15) = q1
β q = 30.
Hence, the original equation is x2 β 13x + 30 = 0
β (x + 10)(x + 3) = 0
β x = -3, -10
Therefore, the roots of the original equation are -3 and -10.
11 and 12 Grade Math
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