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Chapter 8: Problem 57

Solve for \(x\) $$\left|\begin{array}{rr} x+3 & 2 \\ 1 & x+2 \end{array}\right|=0$$

Short Answer

Expert verified
The solutions for \(x\) are -1 and -4.

Step by step solution

01

Calculate the Determinant

The determinant of a 2x2 matrix is calculated using the formula: \(\text{{Det(A)}} = a \cdot d - b \cdot c\), where a, b, c, and d are elements of the matrix. In this case, it will be calculated as: \((x+3) \cdot (x+2) - (1 \cdot 2)\)
02

Set the Determinant Equals to Zero

Since it is given that the determinant equals zero, the equation then becomes: \((x+3) \cdot (x+2) - 2 = 0\)
03

Simplify and Solve for \(x\)

By performing the multiplication and minus operation, the equation simplifies to: \(x^2 + 5x + 6 - 2 = 0\). This further simplifies to \(x^2 + 5x + 4 = 0\). By applying the quadratic formula \(-b \pm \sqrt {b^2 - 4ac} \over 2a\), you find the solutions: \(x = -4\) and \(x = -1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Algebraic Expressions
The process of simplifying algebraic expressions makes them easier to understand and work with. By combining like terms and using the distributive property, expressions can often be reduced to a more manageable form. In the case of our exercise, simplifying the multiplication within the determinant leads to a quadratic equation that can then be solved using the quadratic formula.

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Most popular questions from this chapter

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{x+1}{x^{2}+1}$$

Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form . Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. $$\left[\begin{array}{rrr} 3 & 3 & 3 \\ -1 & 0 & -4 \\ 2 & 4 & -2 \end{array}\right]$$

Determine whether the statement is true or false. Justify your answer. If a system of linear equations has no solution, then the lines must be parallel.

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{c} 2 x+5 y+w=11 \\ x+4 y+2 z-2 w=-7 \\ 2 x-2 y+5 z+w=3 \\ x-3 w=-1 \end{array}\right.$$

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{x^{2}+2}{x^{2}-16}$$
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