Question

Which of the following equations does not have any real solutions?

Answer 1

4x² + 4x + 9 = 0 , has no real solutions

Explanation:

given a quadratic equation in standard form

ax² + bx + c = 0 ( a ≠ 0 )

then the nature of the solutions can be determined using the discriminant.

Δ = b² - 4ac

• if b² - 4ac > 0 , then real solutions

• if b² - 4ac = 0 , then real and equal solutions

• if b² - 4ac < 0 , then no real solutions

4x² + 4x + 9 = 0 ← in standard form

with a = 4, b = 4 , c = 9

b² - 4ac

= 4² - (4 × 4 × 9)

= 16 - 144

= - 128 ← no real solutions

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- 4x² + 56x - 196 = 0 ← in standard form

with a = - 4 , b = 56 , c = - 196

b² - 4ac

= 56² - (4 × - 4 × - 196)

= 3136 - 3136

= 0 ← has real solutions

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- 5x² - 10x + 10 = 0 ← in standard form

with a = - 5 , b = - 10 , c = 10

b² - 4ac

= (- 10)² - (4 × - 5 × 10)

= 100 - (- 200)

= 100 + 200

= 300 ← has real solutions