Math Problem Statement
Find "x":
Solution
The given matrix equation can be expressed as:
[ [x \quad 2 \quad 1] \cdot \begin{bmatrix} 4 & 4 & -4 \ 4 & -1 & 0 \ -4 & -18 & 2 \end{bmatrix} \cdot \begin{bmatrix} x \ -1 \ -4 \end{bmatrix} = [0] ]
We need to compute this matrix multiplication to determine the value(s) of .
Step 1: Multiply the first row vector by the 3x3 matrix.
Performing the first matrix multiplication:
This gives the resulting row vector: Simplifying each element:
Step 2: Multiply the resulting row vector by the column vector.
Now, perform the multiplication of the row vector by the column vector :
Now simplify each term:
Step 3: Combine all the terms.
Now, add all the terms together:
Set the equation equal to 0:
Step 4: Solve the quadratic equation.
We can divide the entire equation by 2 to simplify:
Now, solve this quadratic equation using the quadratic formula: where , , and .
First, calculate the discriminant:
Since the discriminant is negative, there are no real solutions. Therefore, the value of is complex.
The solutions are: Simplifying:
Thus, the solutions are:
Would you like further details on any part of the solution?
Follow-up questions:
- How do you simplify the quadratic equation further?
- What is the significance of a negative discriminant in a quadratic equation?
- How can you confirm if your solution is correct for complex numbers?
- What are the properties of complex roots in quadratic equations?
- Could there be a scenario where has real values for this kind of matrix equation?
Tip: Always check for real solutions before calculating the complex roots in quadratic equations.
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Algebra
Quadratic Equations
Complex Numbers
Formulas
Matrix multiplication formula: C = AB
Quadratic equation formula: ax^2 + bx + c = 0
Quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a
Theorems
Quadratic formula theorem
Complex number solution
Suitable Grade Level
Grades 10-12 (High School)