Question

Without changing their meanings, convert each of the following sentences into a sentence having the form "If \(P\), then \(Q .\) " The discriminant is negative only if the quadratic equation has no real solutions.

Short answer

The converted 'If P, then Q' sentence is: 'If the quadratic equation has no real solutions, then the discriminant is negative.'

Step-by-step solution

Understanding The Provided Sentence

In the sentence 'The discriminant is negative only if the quadratic equation has no real solutions.', there are two main components. The first component or premise is about the discriminant being negative, and the second component or statement is about the quadratic equation having no real solutions. The sentence is essentially implying that if a quadratic equation has no real solutions, then its discriminant is negative.

Formulating The P And Q

In the 'If P, then Q' format, P represents the precondition or the cause, while Q represents the result or the effect. So, for the given sentence, P represents 'the quadratic equation has no real solutions', and Q represents 'The discriminant is negative'.

Transforming The Sentence

Upon identifying P and Q, the original sentence can then be transformed into the 'If P, then Q' format. This leads to: 'If the quadratic equation has no real solutions, then the discriminant is negative.'