Problem 74 - Algebra 1

Question

Use a calculator to solve the equation or write no solution. Round the results to the nearest hundredth. $$\frac{1}{2} x^{2}+3=8$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x\approx 3.16\) and \(x \approx -3.16\)

Step 1: Rearranging the equation

The given equation is \(\frac{1}{2} x^{2}+3=8\). First, we'll subtract 3 from both sides to isolate the term with \(x^{2}\). This gives us \(\frac{1}{2} x^{2} = 8 - 3\), which simplifies to \(\frac{1}{2} x^{2} = 5\).

Step 2: Solve for \(x^{2}\)

Next, to solve for \(x^{2}\), we'll need to multiply both sides of the equation by 2. This gives us \(x^{2} = 5 \times 2\), which simplifies to \(x^{2} = 10\).

Step 3: Solve for \(x\)

Finally, to solve for \(x\), we'll need to take the square root of both sides. Note that when we take the square root of \(x^{2}\), we get two solutions: \(+x\) and \(-x\). Therefore, \(x = \sqrt{10}\) and \(x = -\sqrt{10}\). Use a calculator to find the square roots, rounding to the nearest hundredths, if necessary.

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. They can have zero, one, or two real solutions. To solve them, we can use various methods like factoring, completing the square, or the quadratic formula. In our exercise, the given equation \(\frac{1}{2}x^2 + 3 = 8\) was simplified to \(x^2 = 10\). Solving quadratic equations involves finding the values of \(x\) that satisfy the equation. Once isolated, the next step often involves finding the square roots of the value on the opposite side of the equation.

Rearranging Equations

Rearranging an equation means changing its form without altering the equality, to make it easier to find the unknown. It involves using operations like addition, subtraction, multiplication, or division to both sides.
In the original exercise, the equation \(\frac{1}{2}x^2 + 3 = 8\) was rearranged by subtracting 3 from both sides to isolate the \(x^2\) term, resulting in \(\frac{1}{2}x^2 = 5\).

• Start by moving constant terms to the other side of the equation.
• Then, eliminate coefficients by using the inverse operation.

For example, multiplying both sides by 2 made it easier to solve for \(x^2\). Rearranging is often the first step in solving quadratic or any algebraic equations.

Square Roots

Finding the square root is a crucial step when solving quadratic equations in the form \(x^2 = d\). The square root operation reverses squaring, providing solutions to these equations. It's important to remember that \(x^2\) equations can mean two solutions: both positive and negative values.
Correctly evaluating the square root involves understanding that if \(x^2 = 10\), then \(x = \sqrt{10}\) or \(x = -\sqrt{10}\). In quadratic equations, looking at both the plus and minus roots ensures capturing all possible solutions. Using a calculator will provide the approximate decimal values of these square roots, which are useful when asked to round the answers to a certain decimal place, like in our exercise.

Using a Calculator

Calculators are invaluable tools that help us quickly and accurately perform complex mathematical operations, like finding square roots. When solving equations, inputting calculations correctly ensures the right solutions.
To solve \(x^2 = 10\) using a calculator, start by entering "\(10\)" and then press the square root button. This gives an approximate solution for \(\sqrt{10}\), which should be rounded to two decimal places for precision, resulting in around \(3.16\).

• Always double-check that your calculator is set to the correct mode for the calculation.
• Be careful with negative solutions, remembering to evaluate both \(\sqrt{d}\) and \(-\sqrt{d}\).

Effective calculator use simplifies solving quadratic equations, making it easier to handle roots, coefficients, and more.