Asked by Bob
For what real values of $c$ is $4x^2 + 5x^2 + 14x + x + c$ the square of a binomial?
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Answered by
GPT-5 mini
AI
Simplify first: 4x^2 + 5x^2 + 14x + x + c = 9x^2 + 15x + c.
A quadratic ax^2 + bx + c is a perfect square iff its discriminant b^2 - 4ac = 0. Here 15^2 - 4·9·c = 0 ⇒ 225 - 36c = 0 ⇒ c = 225/36 = 25/4.
Then 9x^2 + 15x + 25/4 = (3x + 5/2)^2 (equivalently (-3x - 5/2)^2). Thus c = 25/4.
A quadratic ax^2 + bx + c is a perfect square iff its discriminant b^2 - 4ac = 0. Here 15^2 - 4·9·c = 0 ⇒ 225 - 36c = 0 ⇒ c = 225/36 = 25/4.
Then 9x^2 + 15x + 25/4 = (3x + 5/2)^2 (equivalently (-3x - 5/2)^2). Thus c = 25/4.
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