Problem 44 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. If the solution involves radicals, round to the nearest hundredth. $$6 n^{2}-10 n+3=0$$

Step-by-Step Solution

Answer

The solutions to the equation are approximately \(n \approx 1.55\) and \(n \approx 0.45\).

Step 1: Identify the coefficients

In the equation \(6n^{2}-10n+3=0\), the coefficient \(a\) is the number before \(n^{2}\) which is \(6\). The coefficient \(b\) is the number before \(n\) which is \(-10\), and the constant \(c\) is \(3\).

Step 2: Substitute the coefficients into the quadratic formula

With \(a=6\), \(b=-10\), and \(c=3\), substitute these values into the quadratic formula \(-b \pm \sqrt{b^{2}-4ac}\)/\(2a\). This yields \(-(-10) \pm \sqrt{(-10)^{2}-4*6*3}\)/\(2*6 = 10 \pm \sqrt{100 - 72}\)/\(12\).

Step 3: Simplify the equation under the square root and the fraction

Continue simplification to get \(10 \pm \sqrt{28}\)/\(12 = 10 \pm 2\sqrt{7}\)/\(12\).

Step 4: Simplify the equation further

The final simplified forms of the solutions will be \(n1 = \frac{5 + \sqrt{7}}{6}\) and \(n2 = \frac{5 - \sqrt{7}}{6}\).

Step 5: Round to the nearest hundredth

Finally, round the solutions to the nearest hundredth: \(n1 \approx 1.55\) and \(n2 \approx 0.45\).

Key Concepts

Coefficients

In any polynomial equation, especially a quadratic equation of the form \(ax^2 + bx + c = 0\), the coefficients are the numbers in front of the variables. Here, they play a crucial role in determining the nature and the values of the solutions. Understanding these coefficients is essential for effectively using the quadratic formula.

• Coefficient \(a\): This is the number multiplied by \(n^2\), which in our equation, \(6n^2 - 10n + 3 = 0\), is 6.
• Coefficient \(b\): The number captured by \(n\), which is \(-10\) in this context.
• Constant \(c\): Unlike \(a\) and \(b\), \(c\) is a standalone number without a variable, which here is 3.

Identifying these coefficients accurately is the first critical step in applying the quadratic formula effectively. Once you have them, you can solve the equation systematically.

Radicals

Radicals, or square roots, arise in calculations when dealing with the quadratic formula, \(-b \pm \sqrt{b^2 - 4ac} \over 2a\). It's essential to recognize and manage radicals properly, as they determine the solution types—real or complex. In our solution:

• We encounter \(\sqrt{28}\), which simplifies to \(2\sqrt{7}\). This process involves recognizing the perfect square factor within 28, which in this case is 4.
• Radicals can be further simplified if the number inside the square root has factors that are perfect squares, which help in expressing the result in its simplest form.

When solving with radicals, always aim to simplify for the clearest and most accurate representation possible.

Solving Quadratic Equations

To solve quadratic equations using the quadratic formula, begin by substituting your identified coefficients into the formula correctly. For the equation \(6n^2 - 10n + 3 = 0\):

• First, identify \(a = 6, b = -10, c = 3\).
• Plug these into the formula to get \(10 \pm \sqrt{100 - 72} \over 12\).

By performing operations under the square root and further division, you arrive at the expression \(10 \pm 2\sqrt{7} \over 12\). Then, simplifying gives \(n1 = \frac{5 + \sqrt{7}}{6}\) and \(n2 = \frac{5 - \sqrt{7}}{6}\).
This process utilizes the principles of algebra, ensuring each step maintains equality and accuracy, leading to the desired solutions.

Rounding Numbers

After arriving at your exact answers using the quadratic formula, rounding becomes the final step if approximations are requested. This often involves rounding to the nearest hundredth for clarity and usability.
In practice, rounding involves:

• Assessing the value in the tenths and hundredths position in your decimal answer.
• The results \(n1 = \frac{5 + \sqrt{7}}{6}\) and \(n2 = \frac{5 - \sqrt{7}}{6}\) approximately equate to 1.55 and 0.45, respectively, after conversion to decimal form using a calculator.
• If the digit after the hundredth place is 5 or greater, round up the hundredths place, otherwise keep it the same.

This ensures the solutions are as accurate as possible while being user-friendly, especially in real-world scenarios where simplified decimals are preferred.