Math Problem Statement
If \( \begin{bmatrix} \frac{a+b}{5} & 2 \\ \frac{1}{ab} & \frac{5}{8} \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & \frac{5}{8} \end{bmatrix} \), then (a, b) is:
Solution
To solve the problem, we have to analyze the given matrix equation:
\frac{a+b}{5} & 2 \\ \frac{1}{ab} & \frac{5}{8} \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & \frac{5}{8} \end{bmatrix}.$$ ### Step-by-Step Solution: 1. **Compare elements in corresponding positions:** - From the first row, first column: $$\frac{a+b}{5} = 6.$$ - From the second row, first column: $$\frac{1}{ab} = 5.$$ 2. **Solve for $$a+b$$:** - Multiply both sides of $$\frac{a+b}{5} = 6$$ by 5: $$a+b = 30.$$ 3. **Solve for $$ab$$:** - Take the reciprocal of $$\frac{1}{ab} = 5$$: $$ab = \frac{1}{5}.$$ 4. **Use the quadratic equation to find $$a$$ and $$b$$:** - Recall the quadratic equation for $$a$$ and $$b$$: $$x^2 - (a+b)x + ab = 0.$$ - Substitute $$a+b = 30$$ and $$ab = \frac{1}{5}$$: $$x^2 - 30x + \frac{1}{5} = 0.$$ 5. **Solve the quadratic equation:** - Multiply through by 5 to eliminate fractions: $$5x^2 - 150x + 1 = 0.$$ - Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$ Here, $$a = 5$$, $$b = -150$$, $$c = 1$$. Substitute: $$x = \frac{-(-150) \pm \sqrt{(-150)^2 - 4(5)(1)}}{2(5)}.$$ Simplify: $$x = \frac{150 \pm \sqrt{22500 - 20}}{10}.$$ $$x = \frac{150 \pm \sqrt{22480}}{10}.$$ 6. **Approximation of roots:** - Calculate $$\sqrt{22480} \approx 149.93$$: $$x = \frac{150 \pm 149.93}{10}.$$ - This gives two solutions: $$x_1 = \frac{150 + 149.93}{10} = 29.993 \quad (\approx 30),$$ $$x_2 = \frac{150 - 149.93}{10} = 0.007 \quad (\approx 0).$$ Therefore, the roots are approximately $$a = 30$$, $$b = 0$$ or vice versa, matching the choice closest to this trend. Do you want more clarification or related problems? Here are five related questions: 1. How can we confirm roots of quadratic equations are accurate in similar scenarios? 2. What alternative methods can be used to cross-check matrices? 3. Explain if such solution roughly format aids calc validations high-impact day? also stack simulation brushed needAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Equality
Quadratic Equations
Formulas
Matrix Equality: Corresponding elements must be equal.
Quadratic Equation Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Theorems
Quadratic Formula
Suitable Grade Level
Grades 11-12