Problem 22 - Algebra 2

Question

Solve each equation by using the Square Root Property. \(x^{2}+12 x+36=5\)

Step-by-Step Solution

Answer

Solutions are \(x = \sqrt{5} - 6\) and \(x = -\sqrt{5} - 6\).

Step 1: Recognize Perfect Square Form

The equation given is \(x^2 + 12x + 36 = 5\). Notice that the left side \(x^2 + 12x + 36\) is a perfect square trinomial because it can be rewritten as \((x + 6)^2\).

Step 2: Rewrite Equation Using Perfect Square

Rewrite the equation \(x^2 + 12x + 36 = 5\) as \((x + 6)^2 = 5\). This is done by recognizing \(x^2 + 12x + 36\) as \((x + 6)^2\).

Step 3: Apply Square Root Property

To solve \((x + 6)^2 = 5\), apply the Square Root Property by taking the square root of both sides, which gives \(x + 6 = \sqrt{5}\) or \(x + 6 = -\sqrt{5}\).

Step 4: Solve for x

Solve the two equations separately. For the first equation, \(x + 6 = \sqrt{5}\), subtract 6 from both sides to get \(x = \sqrt{5} - 6\). For the second equation, \(x + 6 = -\sqrt{5}\), subtract 6 from both sides to get \(x = -\sqrt{5} - 6\). Thus, the solutions are \(x = \sqrt{5} - 6\) and \(x = -\sqrt{5} - 6\).

Key Concepts

Understanding Perfect Square Trinomials

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.

Solving Quadratic Equations Using the Square Root Property

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 for Solving Equations

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.