Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation by extracting square roots. List both the exact answer and a decimal answer that has been rounded to two decimal places. $$ (x+2)^{2}=12 $$

Step-by-Step Solution

Answer

The exact solution is \(x = -2 ± \sqrt{12}\) and the decimal solutions are \(x ≈ 1.46\) and \(x ≈ -5.46\).

Step 1: Simplifying the expression

The equation given is \((x+2)^2 = 12\). The left side of the equation is a perfect square.

Step 2: Apply the square root

Applying the square root to both sides, we have \(x + 2 = ±\sqrt{12}\). However, don't forget that square root of a number gives both positive and negative values.

Step 3: Solving for x

Solving for \(x\) from above, we get \(x = -2 ± \sqrt{12}\). This is our exact solution.

Step 4: Converting to Decimal form

Now, calculating the above to the closest two decimal places, we get that \(x = -2 + \sqrt{12} ≈ 1.46\) and \(x = -2 - \sqrt{12} ≈ -5.46\)

Key Concepts

Square Roots

The concept of square roots is fundamental when working with quadratic equations. Essentially, the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, since \(3 \times 3 = 9\). In a quadratic equation like \((x+2)^2 = 12\), extracting square roots involves finding two values: one positive and one negative, since both \(3^2\) and \((-3)^2\) result in 9.
Applying square roots to both sides of a quadratic equation will help isolate the variable. Remember, when taking the square root on both sides, include the plus-minus \(\pm\) symbol to account for both possibilities.

• Finding the square root of a number \(x\) involves determining values \(y\) such that \(y^2 = x\).
• Symbolically, the square root is represented as \(\sqrt{x}\).
• Always consider both the positive and negative results when solving equations.

Perfect Square

A perfect square is a number that is the square of an integer. Put simply, it's the result of a number multiplied by itself. For instance, 16 is a perfect square because it equals \(4^2\).
In our equation \((x+2)^2=12\), the left side is referred to as a perfect square, since \((x+2)\) multiplied by itself gives \((x+2)^2\). Recognizing perfect squares in equations is a useful trick because it lets you simplify the process of solving for the variable.

• Identifying a perfect square is key in extracting square roots safely.
• Perfect squares simplify the equation-solving process.

Perfect squares should be seen as mathematical shortcuts that streamline resolving quadratics and enhance computational efficiency.

Exact Solution

The exact solution of a quadratic equation is the expression or value that solves the equation without approximation. In the provided problem, the exact solution is obtained directly from the application of the square root on both sides. Since \(x + 2 = \pm \sqrt{12}\), the exact solution becomes \(x = -2 \pm \sqrt{12}\).
Exact solutions do not involve decimal numbers unless those decimals are needed to express precisely rational or irrational numbers as fractions.

• Exact solutions retain mathematical precision.
• These are expressed in roots or radicals for accuracy.

It's important to keep exact solutions in a precise form, as this is often required for deeper mathematical analysis.

Decimal Approximation

Decimal approximation translates an exact solution, often involving roots or radicals, into a more understandable decimal form. This is necessary for practical applications where a precise number helps, for example, estimating or calculating with real-world measurements. In the equation, the exact solution \(x = -2 \pm \sqrt{12}\) translates to decimals \(x \approx 1.46\) and \(x \approx -5.46\) through approximation.
Approximating decimals typically involves rounding off the values to a specific place value, often two decimal places for practical purposes.

• Decimal approximations make solutions more accessible.
• They are easier to interpret in real-world scenarios.
• Always specify the degree of accuracy, such as to the nearest hundredth.

While approximations are useful, bear in mind that they are not exact, and some detail may be lost in translation.