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Chapter 10: Problem 20

Factor. $$x^{2}-12 x+36$$

Short Answer

Expert verified
The factored form of \(x^{2}-12 x+36\) is \((x - 6)^{2}\).

Step by step solution

01

Identify the Perfect Square Trinomial Pattern

Determine whether the given expression \(x^{2}-12x+36\) matches the general form of a perfect square trinomial. The general form is \(a^{2} - 2ab + b^{2}\), which equates to \((a - b)^{2}\). Here, \(a\) corresponds to \(x\), \(2ab\) to \(12x\) and \(b^{2}\) to \(36\). When \(a = x\) and \(b = 6\), the expression perfectly matches the general form of a perfect square trinomial.
02

Factor the Quadratic Expression

Now we have determined the values of \(a\) and \(b\), we can use them to factor the quadratic expression. Taking square root of the first and last term, and taking into account the negative sign of the middle term, we can rewrite the expression as \((x - 6)^{2}\).
03

Verify the Result

Expand \((x - 6)^{2}\) to get back the original quadratic expression, i.e. \(x^{2} - 12x + 36\). This confirms that we have correctly factored it.

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