Problem 53 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation. $$ x^{2}-12 x-3=0 $$

Step-by-Step Solution

Answer

The solutions for the equation \(x^2 - 12x - 3 = 0\) are \(x_1 = 12.541\) and \(x_2 = -0.541\).

Step 1: Identify coefficients

First, you identify the coefficients in the given quadratic equation \(x^2 - 12x - 3 = 0\). In this case, the coefficient \(a\) is 1, \(b\) is -12, and \(c\) is -3.

Step 2: Find the discriminant

Next, calculate the discriminant using the formula \(D = b^2 - 4ac\). Substituting the values of \(a\), \(b\), and \(c\), we get \(D = (-12)^2 - 4*1*(-3) = 144 + 12 = 156.\)

Step 3: Apply the quadratic formula

Now, apply the quadratic formula, \(x = [-b \pm sqrt(D)] / 2a\), substituting values of \(a\), \(b\), and \(D\), gives two solutions for \(x\), namely \(x_1 = [12 + sqrt(156)] / 2 = 12.541\), and \(x_2 = [12 - sqrt(156)] / 2 = -0.541.\)

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool for finding the roots of a quadratic equation, which is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are known values and \( a \) is not zero. This formula reads \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \).

Using this formula, no matter the values of \( a \), \( b \), and \( c \), one can always find the values of \( x \) that satisfy the equation. The \( \pm \) sign indicates that there will generally be two solutions: one for when the square root is added to \( -b \), and another for when it's subtracted. These two solutions are often referred to as the roots or zeros of the quadratic equation.

Real-World Application

One practical application of the quadratic formula is in physics, where it helps in calculating the trajectory of projectiles. Moreover, understanding this formula enhances problem-solving skills and equips students with crucial analytical abilities for various academic fields.

Discriminant of Quadratic Equations

The discriminant of a quadratic equation offers valuable information about the nature of the roots the equation will have without needing to actually calculate them. Mathematically represented as \( D \) and found using the formula \( D = b^2 - 4ac \), it helps us to determine whether the roots are real or complex, and if real, whether they are distinct or repeated.

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one real root (also called a repeated or double root).
• If \( D < 0 \), the equation has two complex roots.

This concept not only assists in solving quadratic equations but is also fundamental in graphing parabolas as it signals the number of intersections with the x-axis.

Significance in Other Fields

Beyond algebra, the discriminant plays a role in fields such as economics, where it can be used to determine equilibrium points, or in engineering to establish the stability of systems.

Coefficients in Algebra

In algebra, coefficients are the numerical or constant part of the terms that are multiplied by the variables. In a quadratic equation of the form \( ax^2 + bx + c = 0 \), \( a \), \( b \), and \( c \) are the coefficients. The value of \( a \) can't be zero because if it were, the equation would no longer be quadratic but linear.

Identifying these coefficients is the critical first step in solving the equation as they are directly used in calculating the discriminant and applying the quadratic formula. Understanding coefficients has practical importances, such as:

• In discussions of motion, \( a \), \( b \), and \( c \) can represent acceleration, initial velocity, and initial position, respectively.
• In economics, they can represent fixed costs, variable costs, and profit when analyzing profit functions.

Recognizing how coefficients impact the shape and position of a parabola is crucial not only for solving equations but also for interpreting graphical data in scientific studies.