Question

The value of p for equation 2x2 – 4x + p = 0 to have real roots will be (a) p ≤ –2 (b) p ≥ 2 (c) p ≤ 2 (d) p ≥ –2

Answer 1

Answer :

(c) p ≤ 2

Note :

★ The possible values of the variable which satisfy the equation are called its roots or solutions .

★ A quadratic equation can have atmost two roots .

★ The general form of a quadratic equation is given as ; ax² + bx + c = 0

★ If α and ß are the roots of the quadratic equation ax² + bx + c = 0 , then ;

• Sum of roots , (α + ß) = -b/a

• Product of roots , (αß) = c/a

★ If α and ß are the roots of a quadratic equation , then that quadratic equation is given as : k•[ x² - (α + ß)x + αß ] = 0 , k ≠ 0.

★ The discriminant , D of the quadratic equation ax² + bx + c = 0 is given by ;

D = b² - 4ac

★ If D = 0 , then the roots are real and equal .

★ If D > 0 , then the roots are real and distinct .

★ If D < 0 , then the roots are unreal (imaginary) .

Solution :

Here ,

The given quadratic equation is ;

2x² - 4x + p = 0 .

Comparing the given quadratic equation with the general quadratic equation ax² + bx + c = 0 ,

we get ;

a = 2

b = -4

c = p

For real roots , the discriminant of the given quadratic equation must be great than or equal to zero .

Thus ,

=> D ≥ 0

=> b² - 4ab ≥ 0

=> (-4)² - 4×2×p ≥ 0

=> 16 - 8p ≥ 0

=> 16 ≥ 8p

=> 8p ≤ 16

=> p ≤ 16/8

=> p ≤ 2

Hence ,

Required answer is :option(c) p ≤ 2 .

Answer 2

Answer:

answer is c p<2

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