Problem 82 - Precalculus with Limits

Question

In Exercises 77-84, solve for \(x\). \(\left| \begin{array}{c} x-2 & -1 \\ -3 & x \end{array} \right| = 0\)

Step-by-Step Solution

Answer

The solutions for \(x\) are 3 and -1.

Step 1: Write Down the Determinant Formula

The determinant formula for a 2x2 matrix is \( |A| = a*d - b*c \). Here a, b, c, d are elements of the matrix A in the order: \(\left| \begin{array}{c} a & b \ c & d \end{array} \right|\). This means a = x - 2, b = -1, c = -3, d = x.

Step 2: Substitute into the Formula

Substituting these values into the determinant formula, we get (x - 2)*x - (-1)*(-3). Simplified, this becomes \(x^2 - 2x - 3 = 0\). This equation equal to 0 gives us the values of \(x\).

Step 3: Solve for \(x\)

Now, we solve the equation from step 2, which is a quadratic equation. In general, a quadratic equation can be solved either by factoring, using the quadratic formula, or completing the square. Here it is more convenient to solve it by factoring. The factors of -3 that add up to -2 are -3 and 1. So, the factored equation becomes \( (x - 3)(x + 1) = 0 \). Then, setting each factor equal to 0, we get the solutions x = 3 or x = -1.

Key Concepts

Understanding the 2x2 Matrix Determinant

The determinant is a special number that can be calculated from a square matrix. In a 2x2 matrix, the determinant provides a simple value that reflects certain properties of the matrix. To calculate the determinant of a 2x2 matrix, we use the formula:

• For a matrix given by \[ \begin{pmatrix} a & b \ c & d \end{pmatrix} \],
• The determinant \(|A|\) is calculated as \( a \cdot d - b \cdot c \).

It's essential to substitute the appropriate matrix elements into this formula accurately. The determinant can help in various applications, including solving systems of linear equations, and determining invertibility of the matrix. If this determinant is zero, as in the exercise above, it indicates that the matrix is singular, which means it doesn't have an inverse.

Factoring Quadratic Equations

Factoring quadratic equations is a method of expressing the equation as a product of two binomials. It’s one of the most efficient ways to find the roots of a quadratic equation when the equation is factorable. A quadratic equation typically looks like this:

• Standard form: \( ax^2 + bx + c = 0 \)

The goal in factoring is to express the equation as \( (px + q)(rx + s) = 0 \).
Here, \(p\) and \(r\) are factors of \(a\), and \(q\) and \(s\) are factors of \(c\), such that their products give the middle term (\(bx\)).
Once in factored form, setting each binomial equal to zero gives the solutions for \(x\).
For instance, in the exercise, the equation\(x^2 - 2x - 3 = 0\)factors into\((x - 3)(x + 1) = 0\),
which gives solutions \(x = 3\) or \(x = -1\). Factoring involves identifying these key values and verifying they satisfy the original equation.

Steps for Solving Quadratic Equations

Solving quadratic equations systematically includes several steps, depending on which method you are using. Here, we'll discuss a few common methods and steps:

• Factoring Method: This involves writing the equation in its factored form and setting each factor equal to zero to solve for \(x\).
• Quadratic Formula: When factoring isn't straightforward, the quadratic formula can be used:\[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]This formula provides a solution for any quadratic equation, without factoring.
• Completing the Square: This method rewrites the quadratic in the form of \((x - p)^2 = q\), providing a different pathway to find \(x\).

In our task at hand, the simplest method was to factor and solve. However, understanding all methods provides a toolbox for tackling any quadratic equation you might encounter in homework or exams.