Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation using any convenient method. \(144-73 x+4 x^{2}=0\)

Step-by-Step Solution

Answer

The solutions for the quadratic equation are \(x = 16\) and \(x = 2.25\).

Step 1: Identify the coefficients

For this quadratic equation \(4x^2 - 73x + 144 = 0\), identify the coefficients for the quadratic formula. Here, \(a = 4\), \(b = -73\), and \(c = 144\).

Step 2: Calculate the discriminant

The discriminant in the quadratic formula is found inside the square root--\(b^2 - 4ac\). Calculate this using the identified coefficients which gives \(b^2 - 4ac = (-73)^2 - 4 * 4 * 144 = 5329 - 2304 = 3025.\)

Step 3: Apply the Quadratic Formula

Plug these values into the quadratic formula. So, \(x = [-(-73) ± sqrt(3025)] / (2*4)\) simplifies to \(x = (73 ± 55)/8\).

Step 4: Solve for x

This will provide two solutions: \(x = (73+55)/8 = 16\) or \(x = (73-55)/8 = 2.25\).

Key Concepts

Quadratic Formula

The quadratic formula is a cornerstone method for solving quadratic equations, which are in the form \( ax^2 + bx + c = 0 \). With the coefficients identified—\( a \), \( b \), and \( c \)—the quadratic formula is: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]. When applied, this formula systematically finds the roots of the quadratic equation, which are the values of \( x \) that make the equation true.

Using the quadratic formula can seem daunting at first, but it becomes straightforward when you break down the process and follow each step carefully. It's important to solve the equation accurately by ensuring that every sign and the coefficient is placed correctly. The \( \pm \) symbol indicates that there will be two solutions, corresponding to both the addition and subtraction of the square root term.

Discriminant Calculation

The discriminant is a key feature of the quadratic formula which tells us about the nature of the roots before even calculating them. It is represented by the expression \( b^2 - 4ac \) inside the square root of the quadratic formula. The discriminant can reveal whether the roots are real or complex, and whether they are distinct or repeated.

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is one real repeated root.
• If it is negative, the roots are complex and come in conjugate pairs.

The discriminant calculation is a straightforward process, but one must ensure all the coefficients are correctly identified and squared or multiplied as per the formula, keeping the sign in mind. In this exercise, a positive discriminant \( 3025 \) indicates that there are two real, distinct solutions.

Quadratic Equation Coefficients

Understanding the role of coefficients in a quadratic equation is fundamental to solving the equation accurately. Coefficients are the numerical parts of the terms that include the variable \( x \). In a standard quadratic equation \( ax^2 + bx + c = 0 \), \( a \) is the coefficient of \( x^2 \) (quadratic term), \( b \) is the coefficient of \( x \) (linear term), and \( c \) is the constant term.

When identifying coefficients, it's crucial to pay attention to their signs. A common mistake is to overlook a negative sign in front of a coefficient, leading to errors in further calculations. Remember to always carry the sign with the coefficient when plugging it into formulas. In the provided exercise, the coefficients are accurately identified as \( a = 4 \), \( b = -73 \) (note the negative sign), and \( c = 144 \) — setting the stage for successful application of the quadratic formula.