Problem 28 - Elementary Algebra

Question

Solve the following quadratic equations. \(n^{2}+8 n+16=27\)

Step-by-Step Solution

Answer

The solutions are \(n = -4 + 3\sqrt{3}\) and \(n = -4 - 3\sqrt{3}\).

Step 1 - Move the constant to the left side

Subtract 27 from both sides of the equation to set the quadratic equation to zero: \(n^{2} + 8n + 16 - 27 = 0\)

Step 2 - Simplify the equation

Combine like terms on the left side: \(n^{2} + 8n - 11 = 0\)

Step 3 - Factor the quadratic equation

Since the equation \(n^{2} + 8n - 11 = 0\) does not factorize easily, we will use the quadratic formula: \[n = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\] Here, \(a = 1\), \(b = 8\), and \(c = -11\).

Step 4 - Apply the quadratic formula

Substitute \(a\), \(b\), and \(c\) into the quadratic formula: \[n = \frac{-8 \pm \sqrt{8^{2} - 4(1)(-11)}}{2(1)}\] Simplify under the square root: \[n = \frac{-8 \pm \sqrt{64 + 44}}{2}\] \[n = \frac{-8 \pm \sqrt{108}}{2}\]

Step 5 - Simplify the square root

Simplify \(\sqrt{108}\): \[n = \frac{-8 \pm \sqrt{36 \cdot 3}}{2}\] \[n = \frac{-8 \pm 6\sqrt{3}}{2}\]

Step 6 - Simplify the expression

Divide each term by 2: \[n = -4 \pm 3\sqrt{3}\] Therefore, the solutions are: \[n = -4 + 3\sqrt{3}\] and \[n = -4 - 3\sqrt{3}\]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are in the form: \[ax^2 + bx + c = 0\]. When factoring is difficult, this formula always works. The quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Here’s what each part means:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

The symbol \(\pm\) means you will get two answers: one using + and one using –. Begin by identifying \(a\), \(b,\) and \(c\), then plug them into the formula. For our example, \(a = 1\), \(b = 8\), and \(c = -11\). Plugging these into the formula gives:\[n = \frac{-8 \pm \sqrt{8^2 - 4(1)(-11)}}{2(1)}\].

Factoring Quadratics

Factoring is another method to solve quadratic equations, and involves rewriting the equation as a product of two binomials. It works best when the quadratic can be easily factored. Consider a simple quadratic equation like \(n^2 + 8n = 20\). The steps are:

• Move all terms to one side to set the equation to zero: \(n^2 + 8n - 20 = 0\)
• Find two numbers that multiply to the constant term, \(-20\), and add to the middle term, \(8\)
• In this case, no easy factor pairs work, but for simpler examples like \(n^2 - 4n - 21 = 0\), we'd use \(n^2 - 7n + 3n - 21 = 0\)
• Then factor by grouping: \((n - 7)(n + 3) = 0\), giving solutions \([n = 7, n = -3]\).

Simplifying Expressions

Simplifying expressions involves combining terms and performing arithmetic to make an expression simpler. For the quadratic formula, simplifying under the square root and afterward is critical. Using our equation \(n = \frac{-8 \pm \sqrt{64 + 44}}{2}\), we proceed as follows:

• Simplify inside the square root: \(\sqrt{108}\)
• Express that as \(\sqrt{36 \cdot 3}\), which is \(6\sqrt{3}\)
• Simplify the whole fraction \(n = \frac{-8 \pm 6\sqrt{3}}{2}\)
• Divide each term: \(n = -4 \pm 3\sqrt{3}\)

Simplify your expressions step by step to avoid mistakes and get clear results.

Combining Like Terms

Combining like terms is a key part of simplifying algebraic expressions. It means trouping terms with the same variable part. For the initial equation we had, \(n^2 + 8n + 16 - 27 = 0\), like terms are combined:

• Combine constants: \(16 - 27 = -11\)
• Result in: \(n^2 + 8n - 11 = 0\)

Remember:

• \(a^2\) and \(a\) are not like terms
• Combine constants like 3 and 5
• Combine variables with the same power, like \(3n\) and \(5n\)

These steps ensure equations are in their simplest form before solving.