Problem 63 - Elementary Algebra

Question

For the following problems, round each result to the nearest hundredth. \(8 a^{2}-168=0\)

Step-by-Step Solution

Answer

Answer: The values of $a$ are approximately $2.60$ and $-2.60$.

Step 1: Identify the coefficients

The given equation is \(8a^2 - 168 = 0\). Comparing it with the general form of a quadratic equation \(Ax^2 + Bx + C = 0\), we can identify the coefficients as: \(A = 8\), \(B = 0\), and \(C = -168\).

Step 2: Plug coefficients into the quadratic formula

Now, we will plug the identified coefficients into the quadratic formula \(a = \frac{-B\pm\sqrt{B^2-4AC}}{2A}\). When we substitute the values, we get: \(a = \frac{-(0)\pm\sqrt{(0)^2 - 4(8)(-168)}}{2(8)}\).

Step 3: Simplify the equation

Simplify the equation by performing the operations inside the square root first, then taking the square root, and finally dividing by the denominator: \(a = \frac{\pm\sqrt{4(8)(168)}}{16}\).

Step 4: Calculate and round the final result

Now calculate the values of the square root, and then divide by \(16\) to obtain the two values for \(a\): \(a_1 = \frac{\sqrt{4(8)(168)}}{16} \approx 2.60\) and \(a_2 = \frac{-\sqrt{4(8)(168)}}{16} \approx -2.60\) Both values of \(a_1\) and \(a_2\) are rounded to the nearest hundredth. The solutions for the given equation are \(a \approx 2.60\) and \(a \approx -2.60\).

Key Concepts

Quadratic Formula

The quadratic formula is a crucial tool for solving quadratic equations, which are polynomials of the form \(Ax^2 + Bx + C = 0\). This standard form allows us to plug in the coefficients into the formula, enabling us to find the roots set by the equation. The formula itself is written as:\[a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\]This formula serves as a universal method for finding solutions to quadratic equations, as long as the coefficients \(A\), \(B\), and \(C\) are known. By taking both the plus and minus of the square root, the formula accounts for the two potential solutions these equations can have.

• The term \(B^2 - 4AC\) under the square root is known as the discriminant. It provides insight into the nature of the solutions, telling us if they are real or complex.
• Practically, after identifying the coefficients, these values are substituted into the formula for resolution.
• It's an efficient method allowing us to bypass more time-consuming graphing or factoring when appropriate.

Rounding numbers

Rounding numbers is the process of reducing the digits in a number while trying to keep its value close to the original. In solving quadratic equations, rounding helps when an exact, clean decimal is beneficial. We aim for simplicity, especially in contexts like engineering or statistical data.In our given exercise, after calculating the potential solutions using the quadratic formula, we round these values to the nearest hundredth. This is typical because working with simpler numbers can be more practical in further calculations or when interpreting results in real-world situations.The steps to round a number to the nearest hundredth are:

• Identify the digit in the hundredths place.
• Look at the digit immediately to its right, the thousandths place.
• If this digit is 5 or more, round up the hundredths digit by one.
• If it's less than 5, leave the hundredths digit unchanged.

Applied to our solution: The acquired roots \(2.598\) and \(-2.598\) approximate to \(2.60\) and \(-2.60\) respectively, providing cleaner numbers to work with.

Coefficient Identification

Before applying the quadratic formula, it's essential to correctly identify the coefficients of the quadratic equation. These coefficients \(A\), \(B\), and \(C\) guide our entire solving process as they determine how we substitute into the formula. In our worked example, the quadratic equation given is \(8a^2 - 168 = 0\). To start the process correctly, compare your equation against the general quadratic form \(Ax^2 + Bx + C = 0\). From here:

• \(A\) represents the coefficient of the \(x^2\) term, which in our exercise is \(8\).
• \(B\) corresponds to the coefficient of the \(x\) term. In this scenario, there is no \(x\) term, so \(B = 0\).
• \(C\) is the constant term, which is \(-168\) here.

Properly identifying these coefficients ensures accuracy when putting values into the quadratic formula, leading to correct solutions of the equation.