Question

The number of integral values of \(x\) for which the determinant of the matrix \(\left[\begin{array}{cc}5 x+14 & -2 \\ 7 x+8 & x\end{array}\right]\) is always less than 1 is (1) 3 (2) 4 (3) 5 (4) 6

Short answer

Answer: 2.

Step-by-step solution

Calculate the determinant

To calculate the determinant of a 2x2 matrix, we can use the formula: \(a d - b c\), where a, b, c, and d are the elements of the matrix in one of the following orders: first row from left to right (a and b) and second row from left to right (c and d). For the given matrix, the determinant can be calculated as follows: \((5x + 14)(x) - (-2)(7x + 8)\).

Determine the inequality

We are given that the determinant is always less than 1. So, we have the inequality: \((5x + 14)(x) - (-2)(7x + 8) < 1\). Now, we need to simplify this inequality to find the bounds on the values of \(x\). We can expand, combine like terms, and obtain a simple quadratic inequality: \((5x + 14)(x) + 2(7x + 8) < 1 \Rightarrow 5x^2 + 14x + 14x + 28 < 1 \Rightarrow 5x^2 + 28x + 27 < 0\).

Find the integral values of \(x\)

To find the integral values of \(x\) that satisfy the quadratic inequality, we can first find the two real roots (if any) of the quadratic equation. The quadratic equation formed by the inequality is: \(5x^2 + 28x + 27 = 0\). Using the quadratic formula, we can find the real roots: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-28 \pm \sqrt{28^2 - 4(5)(27)}}{10}\). Calculating the roots, we find that they are real and are approximately equal to \(-1.433\) and \(-3.767\). Since the inequality is less than 0, the integral values of \(x\) that satisfy the inequality are between these two roots (not inclusive). Hence, we can conclude that there are 2 integral values of \(x\) (those are -2 and -3) that satisfy the inequality \(5x^2 + 28x + 27 < 1\). Therefore, the answer is: (1) 3

Key concepts

Determinants of Matrices

Determinants are a significant property of matrices used to understand various matrix attributes. For a 2x2 matrix, the determinant is calculated simply using the formula: \( ad - bc \), where \( a \), \( b \), \( c \), and \( d \) are elements of the matrix given as follows: \( \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \).
For the matrix in the example, denoted as: \( \left[\begin{array}{cc}5x+14 & -2 \ 7x+8 & x \end{array}\right] \), the determinant is calculated by substituting \( a = 5x + 14 \), \( b = -2 \), \( c = 7x + 8 \), and \( d = x \).
Thus, the determinant becomes: \( (5x + 14)x - (-2)(7x + 8) \). This simplifies to \( 5x^2 + 14x + 14x + 28 \), giving the quadratic expression \( 5x^2 + 28x + 28 \).
Determinants help in discovering various solutions, and in this context, they assist in framing inequalities that define possible values for \( x \). Understanding determinants is crucial for tackling problems that involve conditions or restrictions set by matrix operations.

Integral Solutions

Finding integral solutions involves identifying values that satisfy given conditions, with the additional constraint that these values must be whole numbers. In the exercise, our focus is on determining which integral values of \( x \) fit the requirement of the determinant being less than 1.
The key is to solve the quadratic inequality \( 5x^2 + 28x + 27 < 0 \), which involves finding the range of \( x \). Solutions between the roots of the related quadratic equation will inform where the inequality holds true.
Once the roots are approximated as \(-1.433\) and \(-3.767\), any whole number solutions within this range must be checked individually.

• The integer values \(-2\) and \(-3\) clearly fit these criteria.
• Verifying these satisfies our original inequality: substitute and check that the inequality holds true.

Thus, careful evaluation and testing secure the correct integral solutions, providing insight that is often necessary in algebraic problem-solving.

Quadratic Formula

The quadratic formula provides a method to find solutions for quadratic equations of any form \( ax^2 + bx + c = 0 \). The formula is given as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This tool is particularly powerful because it works regardless of whether the roots are rational, irrational, or complex.
In our exercise, for the equation \( 5x^2 + 28x + 27 = 0 \), the coefficients are \( a = 5 \), \( b = 28 \), and \( c = 27 \). By substituting these into the quadratic formula, we compute the discriminant \( b^2 - 4ac = 28^2 - 4 \times 5 \times 27 \) first.
Calculating it yields positive results, indicating real roots. Solving further, you find the roots through:

• First root: \(-1.433\)
• Second root: \(-3.767\)

Knowing how to apply this formula enhances problem-solving capability when dealing with quadratic equations, providing a gateway to understanding inequalities and real-world applications.

Roots of Quadratic Equations

Roots are the solutions to quadratic equations that satisfy \( ax^2 + bx + c = 0 \). Each root represents where the equation equals zero, playing a critical role in problem-solving.
In the presented exercise, after implementing the quadratic formula, we identified roots \(-1.433\) and \(-3.767\).
Understanding these roots helps us discern regions where the inequality \( 5x^2 + 28x + 27 < 0 \) holds true.

• Any values of \( x \) lying between these roots satisfy the inequality condition.
• Checking integer solutions shows that only \(-2\) and \(-3\) work.

This understanding of locating and interpreting roots helps solve various algebraic and calculus problems. When roots are clearly at hand, you can map out solution sets, graph functions, and solve equations efficiently. Such knowledge is invaluable for solving complex inequalities and understanding the behavior of quadratic equations.