Problem 55 - Contemporary Precalculus

Question

Use a calculator to find approximate solutions of the equation. $$3 x^{2}-82.74 x+570.4923=0$$

Step-by-Step Solution

Answer

Answer: The approximate solutions are x_1 ≈ 12.0272 and x_2 ≈ 15.6731.

Step 1: Identify the coefficients

In the given equation, the coefficients are: a = 3, b = -82.74, c = 570.4923

Step 2: Calculate the discriminant

The discriminant (Δ) can be calculated by using the formula: $$Δ = b^2 - 4ac$$ Substitute the values of a, b, and c in the formula: $$Δ = (-82.74)^2 - 4(3)(570.4923)$$ Use a calculator to find the value of Δ: $$Δ ≈ 3354.6452$$

Step 3: Apply the quadratic formula

The quadratic formula for finding the roots of the equation is given by: $$x = \frac{-b \pm \sqrt{Δ}}{2a}$$ Substitute the calculated values of a, b, and Δ in the formula: $$x = \frac{82.74 \pm \sqrt{3354.6452}}{2(3)}$$ Now, calculate the 2 different x-values ("+" and "-") separately using the formula:

Step 4: Calculate x value 1

Calculate the value when we use the "+" sign in the formula: $$x_1 = \frac{82.74 + \sqrt{3354.6452}}{6}$$ $$x_1 ≈ 12.0272$$

Step 5: Calculate x value 2

Calculate the value when we use the "-" sign in the formula: $$x_2 = \frac{82.74 - \sqrt{3354.6452}}{6}$$ $$x_2 ≈ 15.6731$$ The approximate solutions of the given quadratic equation are approximately x_1 ≈ 12.0272, and x_2 ≈ 15.6731.

Key Concepts

Quadratic Formula

Understanding the quadratic formula is crucial for solving quadratic equations. Quadratic equations have the general form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \eq 0\). The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) offers a straightforward method for finding the roots of any quadratic equation. Here's why each component is important:

• The \(b\) term represents the linear coefficient, which can profoundly influence the position of the parabola on a graph.
• \(a\), the leading coefficient, determines the parabola's width and direction – opening up if positive and down if negative.
• \(c\), the constant term, dictates the y-intercept of the parabola.

The formula consists of two parts, the negative/positive sign in \(\pm\), which means there can be two solutions for \(x\): one from adding the square root, and the other from subtracting, leading to potentially two distinct intersections with the \(x\)-axis. Always ensure to include both roots in the final solution to cover all possible points where the parabola crosses or touches the \(x\)-axis.

Discriminant

The discriminant is the part under the square root in the quadratic formula, denoted as \(\Delta = b^2 - 4ac\). It is a powerful tool that tells us about the nature of the roots without solving the equation entirely.

• If \(\Delta > 0\), there are two real and distinct solutions, meaning the graph of the parabola crosses the \(x\)-axis at two points.
• If \(\Delta = 0\), there is exactly one real solution, which implies that the parabola touches the \(x\)-axis at a single point, known as a \

Calculator Use in Algebra

Calculators play a pivotal role in algebra, especially when dealing with complex numerical values. They assist students in carrying out operations that would be cumbersome by hand, such as finding square roots or handling decimals and large numbers. When using a calculator, it's essential to follow a few best practices:

• Double-check that you've entered the coefficients correctly, especially signs (positive/negative).
• Use parentheses to group terms accurately, ensuring the order of operations is maintained.
• Be familiar with your calculator's functions, such as the square root and exponentiation keys.

When working with the discriminant and the quadratic formula, a calculator can help verify your results quickly, ensuring that you're on the right track before proceeding further. However, understanding the underlying algebraic principles is crucial; relying solely on technology might hinder developing essential problem-solving skills. Always complement calculator use with algebraic reasoning.