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A perfect square trinomial is a special type of quadratic expression that can be factored into a square of a binomial. It takes the form of

• \(a^2 + 2ab + b^2\)
• this can be rewritten as \((a + b)^2\).

In our given equation, \(x^2 + 12x + 36\), we see it is a perfect square trinomial because it fits this pattern. Let's break this down further. The constant, 36,is a square number (\(6^2 = 36\)), and the middle term, 12x, is twice the product of 6 and x \((2 \times 6 \times x = 12x))\). This tricky little pattern helps us recognize and simplify perfect square trinomials into binomials such as \((x + 6)^2\). By identifying these patterns, solving quadratic equations becomes more manageable.

Quadratic equations are typically in the form \(ax^2 + bx + c = 0\), and solving them is a fundamental skill in algebra. One effective method is using the Square Root Property, particularly beneficial when the expression is a perfect square trinomial. After rewriting \(x^2 + 12x + 36 = 5\) as \((x + 6)^2 = 5\), we simplify the problem. The Square Root Property states that if \(a^2 = b\), then \(a = \sqrt{b}\) or \(a = -\sqrt{b}\). Applying this property helps us directly solve for x by taking the square root of both sides:

• \((x + 6) = \sqrt{5}\)
• or \((x + 6) = -\sqrt{5}\).

This splits the problem into two simpler equations that can be solved for x. This method provides a direct path to finding x and understanding the structure of the solutions.

Algebraic manipulation involves various strategies to isolate the variable and solve the equation. In our exercise, once we reached the equations

• \(x + 6 = \sqrt{5}\)
• and \(x + 6 = -\sqrt{5}\).

We perform further manipulation to find the value of x. Subtracting 6 from both sides of each equation helps isolate x. This gives two distinct solutions: \(x = \sqrt{5} - 6\) and \(x = -\sqrt{5} - 6\). By understanding these algebraic transformations, solving becomes more intuitive. Such skills are invaluable in handling equations in mathematics, emphasizing logic and careful application of steps to arrive at a solution efficiently.