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Chapter 0: Problem 49

Factor each perfect square trinomial. $$x^{2}+2 x+1$$

Short Answer

Expert verified
The factored form of the perfect square trinomial \(x^{2}+2x+1\) is \((x+1)^{2}\)

Step by step solution

01

Identify the Perfect Square Trinomial

Look at the given expression \(x^{2}+2x+1\), which is in the form \(a^{2}+2ab+b^{2}\). In this form, \(a\) and \(b\) are real numbers. Thus, \(a = x\), \(2ab = 2x\), and \(b^{2} = 1\). Based on these identifications, it can be assumed that \(a = x\) and \(b = 1\). This is because \(b^{2} = 1^{2} = 1\) and \(2ab = 2.x.1 = 2x\), which matches the structure of the given trinomial.
02

Factor the Perfect Square Trinomial

Now, express the trinomial \(x^{2}+2x+1\) as the square of a binomial. Based on the previous step, \(a = x\) and \(b = 1\). So, the binomial is \(a + b = x + 1\). Now, square this to obtain the factored form of the perfect square trinomial. So, the factored form of the perfect square trinomial is \((x+1)^{2}\).
03

Verification of the Factored Form

To verify the factored form, expand \((x+1)^{2}\) to check if it corresponds to the original trinomial. Expanding the square, we get \(x^{2} + 2x + 1\), which is the original trinomial. This confirms the factored form is correct.

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What is the discriminant and what information does it provide about a quadratic equation?

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