Problem 88 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. $$ 9 d^{2}-58 d+24=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(9d^{2} - 58d + 24 = 0\) are \(d_1 = 6\) and \(d_2 = \frac{4}{9}\).

Step 1: Identification of coefficients

In order to use the quadratic formula, first identify the coefficients \(a\), \(b\), and \(c\) in the given quadratic equation. For this equation \( 9d^{2} - 58d + 24 = 0 \), \(a\) is 9, \(b\) is -58, and \(c\) is 24.

Step 2: Substitute the coefficients into the quadratic formula

Now substitute the coefficients into the quadratic formula, \(d = \frac{{-b \pm \sqrt{{b^{2}-4ac}}}}{{2a}}\), to get \(d = \frac{{58 \pm \sqrt{{(-58)^{2}-4*9*24}}}}{{2*9}} = \frac{{58 \pm \sqrt{{3364-864}}}}{{18}} = \frac{{58 \pm \sqrt{{2500}}}}{{18}}.\)

Step 3: Solve the equation

Solve the equation to find the values of \(d\). This gives \(d = \frac{{58 \pm 50}}{{18}}\). So the two solutions are \(d_1 = \frac{{108}}{{18}} = 6\) and \(d_2 = \frac{{8}}{{18}} = \frac{{4}}{{9}}\).

Key Concepts

Solving Quadratic Equations

To solve quadratic equations efficiently, one powerful method we use is the quadratic formula. This formula is particularly helpful when factoring is complex or impossible. Quadratic equations are of the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, with \( a eq 0 \). The quadratic formula provides the solutions to these equations directly. These solutions, also known as roots or zeros, can be calculated using:

• \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \)

The expression within the square root, \( b^2 - 4ac \), is called the discriminant. It tells us about the nature of the roots:

• If the discriminant is positive, two distinct real roots exist.
• If zero, one real root (a repeated root) exists.
• If negative, no real roots exist (the solutions are complex).

Understanding and applying the quadratic formula can turn any quadratic equation into a solvable problem, as demonstrated in our example problem involving \( 9d^2 - 58d + 24 = 0 \). By identifying the right coefficients and substituting them into the formula, you can find precise solutions.

Coefficients in Quadratic Equations

In a quadratic equation, the coefficients \( a, b, \), and \( c \) play significant roles. Identifying them correctly is crucial for applying the quadratic formula. Let's break down these coefficients:

• \( a \): The coefficient of the quadratic term (\( x^2 \)). It determines the parabola's direction: upwards if \( a > 0 \) and downwards if \( a < 0 \).
• \( b \): The coefficient of the linear term (\( x \)). This affects the parabola's horizontal position on the graph.
• \( c \): The constant term. It indicates the parabola's vertical position or the y-intercept.

In the equation \( 9d^2 - 58d + 24 = 0 \), \( a = 9 \), \( b = -58 \), and \( c = 24 \). These coefficients are essential when calculating the discriminant and substituting into the quadratic formula. Mistakes in identifying these values can lead to incorrect solutions. Hence, it's a step that mustn't be overlooked.

Steps in Solving Quadratic Equations

Solving a quadratic equation using the quadratic formula involves several key steps. Here they are explained in an approachable way:

Step 1: Identify Coefficients

Begin by recognizing the coefficients \( a \), \( b \), and \( c \) from the equation \( ax^2 + bx + c = 0 \). Replace these into the quadratic formula.

Step 2: Apply the Quadratic Formula

Place these coefficients into the formula: \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \). It's the heart of solving these equations. Calculate the discriminant, \( b^2 - 4ac \), first as it tells you the type of solutions you expect.

Step 3: Solve for Roots

Complete the calculation to find the roots of the equation. Depending on the discriminant's value, you will find these roots:

• Two solutions: \( x_1 \) and \( x_2 \) if the discriminant is positive.
• A single repeated solution if it's zero.
• Complex solutions if negative, although not applicable in this example.

Following these steps, as shown with \( 9d^2 - 58d + 24 = 0 \), ensures a clear path to the correct solution, enhancing understanding and building confidence in algebraic problem-solving.