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The quadratic equation x2 + bx + 4 = 0 will have real roots, if
- a)Only b ≤ –4
- b)Only b ≥ 4
- c)–4 < b < 4
- d)b ≤ –4, b ≥ 4
Correct answer is option 'D'. Can you explain this answer?
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The quadratic equation x2 + bx + 4 = 0 will have real roots, ifa)Only ...
To determine the conditions for the quadratic equation to have real roots, we need to use the discriminant formula. The discriminant is given by b^2-4ac, where a, b and c are the coefficients of the quadratic equation ax^2+bx+c=0.
If the discriminant is positive, then the quadratic equation will have two distinct real roots.
If the discriminant is zero, then the quadratic equation will have one real root (a repeated root).
If the discriminant is negative, then the quadratic equation will have two complex roots.
Using this information, we can solve the given quadratic equation x^2-bx+4=0 as follows:
Discriminant = b^2-4ac
= b^2-4(1)(4)
= b^2-16
For the quadratic equation to have real roots, the discriminant must be greater than or equal to zero.
Therefore, b^2-16 ≥ 0
(b-4)(b+4) ≥ 0
This inequality is satisfied when:
b ≤ -4 or b ≥ 4
Hence, the quadratic equation x^2-bx+4=0 will have real roots if b ≤ -4 or b ≥ 4.
Therefore, the correct answer is option 'D': b ≤ -4 or b ≥ 4.
If the discriminant is positive, then the quadratic equation will have two distinct real roots.
If the discriminant is zero, then the quadratic equation will have one real root (a repeated root).
If the discriminant is negative, then the quadratic equation will have two complex roots.
Using this information, we can solve the given quadratic equation x^2-bx+4=0 as follows:
Discriminant = b^2-4ac
= b^2-4(1)(4)
= b^2-16
For the quadratic equation to have real roots, the discriminant must be greater than or equal to zero.
Therefore, b^2-16 ≥ 0
(b-4)(b+4) ≥ 0
This inequality is satisfied when:
b ≤ -4 or b ≥ 4
Hence, the quadratic equation x^2-bx+4=0 will have real roots if b ≤ -4 or b ≥ 4.
Therefore, the correct answer is option 'D': b ≤ -4 or b ≥ 4.
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