Problem 73 - Intermediate Algebra

Question

The solution set for \(x^{2}-4 x-37=0\) is \(\\{2 \pm \sqrt{41}\\}\). With a calculator, we found a rational approximation, to the nearest one-thousandth, for each of these solutions. $$ 2-\sqrt{41}=-4.403 \quad \text { and } \quad 2+\sqrt{41}=8.403 $$ Thus the solution set is \(\\{-4.403,8.403\\}\), with the answers rounded to the nearest one-thousandth. Solve each of the equations in Problems 64-73, expressing solutions to the nearest one-thousandth. $$ 3 x^{2}-12 x-10=0 $$

Step-by-Step Solution

Answer

The solutions are approximately \(x = 4.208\) and \(x = -0.708\).

Step 1: Identify the Coefficients

In the given quadratic equation \(3x^2 - 12x - 10 = 0\), identify the coefficients as \(a = 3\), \(b = -12\), and \(c = -10\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is calculated using the formula \(b^2 - 4ac\). Substitute the given values: \((-12)^2 - 4(3)(-10)\).

Step 3: Simplify the Discriminant

Calculate the result of the discriminant: \(144 + 120 = 264\).

Step 4: Use the Quadratic Formula

The solutions are given by the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the known values: \(x = \frac{-(-12) \pm \sqrt{264}}{2 \times 3}\).

Step 5: Simplify the Expression

Simplify the equation: \(x = \frac{12 \pm \sqrt{264}}{6}\).

Step 6: Compute the Square Root

Calculate \(\sqrt{264}\) approximately with a calculator: \(\sqrt{264} \approx 16.248\).

Step 7: Solve for the Solutions

Solve \(x = \frac{12 \pm 16.248}{6}\):
- First solution: \(x = \frac{12 + 16.248}{6} \approx 4.208\).
- Second solution: \(x = \frac{12 - 16.248}{6} \approx -0.708\).

Step 8: Round to Nearest Thousandth

Round the solutions to the nearest thousandth: \(x \approx 4.208\) and \(x \approx -0.708\).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool when solving quadratic equations. It can be used to find the roots or solutions given any quadratic equation of the form: \[ ax^2 + bx + c = 0 \]The formula itself is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here is how it works:

• \(a\), \(b\), and \(c\) represent the coefficients of the equation.
• \(b^2 - 4ac\) is known as the discriminant.
• The formula provides two solutions: one with plus \((+)\) and one with minus \((-)(\pm)\). This is due to the two possible roots in many quadratic equations.

This formula works for all quadratic equations, even those that are not easily factorable, and provides an exact answer. Using the quadratic formula ensures a consistent method to find solutions by directly substituting the coefficients of the equation.

Discriminant

The discriminant is an integral part of the quadratic formula that determines the nature of the roots of a quadratic equation. Given by the expression:\[ b^2 - 4ac \]The discriminant helps us understand the type of solutions without solving the full equation. Here's what you need to know:

• If the discriminant is positive, the quadratic equation will have two distinct real roots. This is the case for our example where the discriminant was calculated as 264.
• If it is zero, there will be exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the quadratic equation has no real roots. Instead, the solutions are complex or imaginary numbers.

Thus, the discriminant is a useful tool in predicting the nature of roots before proceeding with detailed calculations.

Rational Approximation

When solving quadratic equations, we sometimes need to approximate irrational numbers. These are numbers that cannot be expressed exactly as fractions. For example, the square root of a non-perfect square, like \(\sqrt{264}\), often results in an irrational number. A rational approximation helps provide a nearby rational number that is easy to work with. In many contexts, especially in solving problems by hand or when using a calculator, you aim to approximate to a certain precision, often to the nearest thousandth or hundredth digit. To achieve a good approximation:

• First, calculate the square root or other expressions to a sufficient level of decimal places using a calculator.
• Then, round to the required number of decimal places. Here we rounded \(\sqrt{264}\) to approximately 16.248, which helped to simplify calculating the final solution values: \(4.208\) and \(-0.708\).

Rational approximations allow us to manage and use otherwise unwieldy irrational numbers in practical applications.

Coefficients in Quadratic Equations

Coefficients in a quadratic equation are the numbers placed before the variables and constant. In the general form of the quadratic equation \(ax^2 + bx + c = 0\), these coefficients determine the curve's shape and position on a graph and are crucial in solving the equation.Here’s a breakdown:

• \(a\) is the coefficient of \(x^2\). It affects the width and direction of the parabola. If \(a\) is positive, the parabola opens upward; if negative, it opens downward.
• \(b\) is the coefficient of \(x\). It influences the horizontal position and the symmetry of the parabola.
• \(c\) is the constant term. It indicates where the parabola intersects the y-axis.

In the context of solving quadratic equations like \(3x^2 - 12x - 10 = 0\), identifying these coefficients correctly is the first step to using the quadratic formula effectively. Correctly plugging in these coefficients into the quadratic formula or calculating the discriminant enables accurate determination of roots, leading to successful solutions.