Quadratic Equation
- The equation x2 − px + q = 0, p, q ∈ R has on real root if :
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According to question , we can say that
The equation x2 − px + q = 0, p, q ∈ R has no real root if B2 < 4ACCorrect Option: B
According to question , we can say that
The equation x2 − px + q = 0, p, q ∈ R has no real root
On comparing with quadratic eq. Ax2 + Bx + C = 0 , we get
∴ A =1, B= - p, C = q
Now ,we have B2 < 4AC
⇒ ( - P )2 < 4 x 1 x q
⇒ p2 < 4q.
Hence , option B is correct answer .
- x = 3a solution of the equation 3x2 + (k − 1)x + 9 = 0 if k has value
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The given quadratic equation 3x2 + (k − 1)x + 9 = 0
Putting x = 3 in above given eq. , we get
27 + 3(k − 1) + 9 = 0
⇒ 27 + 3k - 3 + 9 = 0
⇒ 3k = −33 ⇒ k = −11.Correct Option: D
The given quadratic equation 3x2 + (k − 1)x + 9 = 0
Putting x = 3 in above given eq. , we get
27 + 3(k − 1) + 9 = 0
⇒ 27 + 3k - 3 + 9 = 0
⇒ 3k = −33 ⇒ k = −11.
Hence , the value of k is −11.
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One root of the equation 3x2 - 10x + 3 = 0 is 1 . Find the other root. 3
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The given quadratic equation is 3x2 - 10x + 3 = 0
Comparing with ax2 + bx + c = 0, we get
a = 3, b = −10, c = 3Correct Option: A
The given quadratic equation is 3x2 - 10x + 3 = 0
Comparing with ax2 + bx + c = 0, we get
a = 3, b = −10, c = 3∴ Sum of the roots ( k1 + k2 ) = - a = 10 b 3 ∵ One root k1 = 1 3 ∴ The other root k2 = 10 - 1 = 9 = 3. 3 3 3
Therefore , The other root is 3 .
- The expression x4 + 7x2 + 16 can be factored as :
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The given expression x4 + 7x2 + 16 = ( x4 + 8x2 + 16 ) - x2
= ( x4 + 2 * 4 * x2 + 42 ) - x2
= ( x2 + 4 )2 - x2Correct Option: C
The given expression x4 + 7x2 + 16 = ( x4 + 8x2 + 16 ) - x2
= ( x4 + 2 * 4 * x2 + 42 ) - x2
= ( x2 + 4 )2 - x2
= ( x2 + 4 + x ) ( x2 + 4 - x )
Hence , x4 + 7x2 + 16 = ( x2 + x + 4 ) ( x2 - x + 4 ).
Option C is correct answer .
- The common root of the equations x2 - 7x + 10 = 0 and x2 - 10x + 16 = 0 is :
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As per the given above question , we can say that
The given quadratic equations are
x2 - 7x + 10 = 0 ⇔ (x − 5)(x − 2) = 0
And x2 - 10x + 16 = 0 ⇔ (x − 8)(x − 2) = 0Correct Option: D
As per the given above question , we can say that
The given quadratic equations are
x2 - 7x + 10 = 0 ⇔ (x − 5)(x − 2) = 0 ⇔ x = 5, 2
And x2 - 10x + 16 = 0 ⇔ (x − 8)(x − 2) = 0 ⇔ x = 8, 2
∴ Common root is 2.