Problem 53 - Algebra 1: Concepts and Skills

Question

Use a calculator to solve the equation. Round the result to the nearest hundredth. $$ 3 x^{2}+7=31 $$

Step-by-Step Solution

Answer

The solution to the equation \(3 x^{2}+7=31\) rounded to the nearest hundredth is \(x=2.83\) and \(x=-2.83\)

Step 1: Rearrange the equation

Start by rearranging the equation to be in the form of \(ax^2+bx+c=0\). In this case, subtract 7 from both sides of the equation: \(3x^2+7-7=31-7\), which simplifies to \(3x^2=24\)

Step 2: Isolate the variable

Divide both sides of the equation by 3 to make the leading coefficient 1: \(x^2 = 24/3\), simplifying gives \(x^2=8\)

Step 3: Solve for x

Since the equation is now \(x^2=8\), you simply take the square root of both sides to solve for x. One key point to remember is that solving this way has two potential results - positive and negative. So \(x = \sqrt{8}\) or \(x = -\sqrt{8}\). Calculating this gives a decimal of approximately \(2.83\) and \(-2.83\).

Step 4: Round to the nearest hundredth

Finally, round both roots to the nearest hundredth if necessary. However, in this case no further action is required.

Key Concepts

Solving Quadratic Equations

Quadratic equations have the standard form as follows: \(ax^2 + bx + c = 0\). Solving these equations often requires rearranging or simplifying to isolate the \(x^2\) term.

• In the example exercise, the equation \(3x^2 + 7 = 31\) was simplified by moving constants to one side, resulting in \(3x^2 = 24\).
• The next step involves isolating \(x^2\), leading to dividing by the coefficient of \(x^2\), making \(x^2 = 8\).
• Lastly, apply an operation to both sides of the equation that simplifies \(x\), most commonly the square root.

Quadratic equations can also be solved using methods like factoring, completing the square, or using the quadratic formula, depending on what’s most appropriate for the given equation.

Square Roots

Finding the square root is a method used to determine a number which, when multiplied by itself, gives the original number. The operation is denoted by a radical symbol \(\sqrt{}\). For equations of the form \(x^2 = a\), solving for \(x\) involves taking the square root on both sides:

• In our problem, with \(x^2 = 8\), we calculate \(x = \sqrt{8}\).
• It’s important to note that square roots can produce two values, one positive and one negative—hence \(x = \pm \sqrt{8}\).
• Using a calculator, \(\sqrt{8}\) approximately equals 2.828, but keep both values: \(2.83\) and \(-2.83\) for solutions.

Square roots are invaluable when dealing directly with quadratic equations, helping us find roots quickly and efficiently.

Rounding Decimals

Rounding is a simple numerical method used to approximate a number by maintaining certain precision. It's useful for simplifying long decimal numbers into manageable figures. When rounding to the nearest hundredth:

• Focus on the digit in the hundredth place (second digit after the decimal) and the digit that follows it.
• If the digit after the hundredth place is 5 or more, round the hundredth digit up.
• If it's less than 5, keep the hundredth digit as it is.

In the context of our exercise, the square root \(\sqrt{8} \approx 2.828\) becomes \(2.83\) when rounded to the nearest hundredth. Rounding helps achieve a balance between accuracy and simplicity, especially when solutions will later be used in further computations or comparisons.