if a,b,c are real numbers such that ac is not equal to 0,then show that at least one of the equations ax2 + bx + c = 0 and -ax2 + bx + c = 0 has real roots.

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Solution

Given, ac ≠ 0
Consider the quadratic equation ax ² + bx + c = 0
D₁ = b ² – 4ac (1)
Consider the quadratic equation – ax ² + bx + c = 0
D₂ = b ² + 4ac (2)
Adding (1) and (2), we get
D₁ + D₂ = 2b ² ≥ 0
Hence, at least one of D₁ and D₂ ≥ 0
Therefore, at least one of the two given quadratic equations has real roots.

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