Math Problem Statement

Find "x":

Determine the value(s) of x such that [x 2 1] [4 4 -4; 4 -1 0; -4 -18 2] [x -1 -4] = [0]

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 xx.

Step 1: Multiply the first row vector by the 3x3 matrix.

Performing the first matrix multiplication: [x21][4444104182][x \quad 2 \quad 1] \cdot \begin{bmatrix} 4 & 4 & -4 \\ 4 & -1 & 0 \\ -4 & -18 & 2 \end{bmatrix}

This gives the resulting row vector: [4x+8+4,4x2+0,4x36+2][4x + 8 + -4, 4x - 2 + 0, -4x - 36 + 2] Simplifying each element: [4x+4,4x2,4x34][4x + 4, 4x - 2, -4x - 34]

Step 2: Multiply the resulting row vector by the column vector.

Now, perform the multiplication of the row vector [4x+4,4x2,4x34][4x + 4, 4x - 2, -4x - 34] by the column vector [x14]\begin{bmatrix} x \\ -1 \\ -4 \end{bmatrix}: (4x+4)x+(4x2)(1)+(4x34)(4)(4x + 4)x + (4x - 2)(-1) + (-4x - 34)(-4)

Now simplify each term: (4x+4)x=4x2+4x(4x + 4)x = 4x^2 + 4x (4x2)(1)=4x+2(4x - 2)(-1) = -4x + 2 (4x34)(4)=16x+136(-4x - 34)(-4) = 16x + 136

Step 3: Combine all the terms.

Now, add all the terms together: 4x2+4x4x+2+16x+136=4x2+16x+1384x^2 + 4x - 4x + 2 + 16x + 136 = 4x^2 + 16x + 138

Set the equation equal to 0: 4x2+16x+138=04x^2 + 16x + 138 = 0

Step 4: Solve the quadratic equation.

We can divide the entire equation by 2 to simplify: 2x2+8x+69=02x^2 + 8x + 69 = 0

Now, solve this quadratic equation using the quadratic formula: x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where a=2a = 2, b=8b = 8, and c=69c = 69.

First, calculate the discriminant: Δ=b24ac=824(2)(69)=64552=488\Delta = b^2 - 4ac = 8^2 - 4(2)(69) = 64 - 552 = -488

Since the discriminant is negative, there are no real solutions. Therefore, the value of xx is complex.

The solutions are: x=8±4884=8±488i4x = \frac{-8 \pm \sqrt{-488}}{4} = \frac{-8 \pm \sqrt{488}i}{4} Simplifying: x=8±22.09i4x = \frac{-8 \pm 22.09i}{4} x=2±5.52ix = -2 \pm 5.52i

Thus, the solutions are: x=2+5.52i,x=25.52ix = -2 + 5.52i, \quad x = -2 - 5.52i

Would you like further details on any part of the solution?

Follow-up questions:

  1. How do you simplify the quadratic equation further?
  2. What is the significance of a negative discriminant in a quadratic equation?
  3. How can you confirm if your solution is correct for complex numbers?
  4. What are the properties of complex roots in quadratic equations?
  5. Could there be a scenario where xx 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)