Example Question - root finding
Here are examples of questions we've helped users solve.
Solving an Equation with Square Root
Bài toán trong hình đặt ra là: Giải phương trình: \(x^2 - 3x + 2 - \sqrt{x - 1} = x^2 - 3x + 2\) Để giải phương trình này, trước hết ta xác định điều kiện xác định của phương trình: \(\sqrt{x - 1}\) tồn tại \( \rightarrow x - 1 \geq 0 \rightarrow x \geq 1\). Do hai vế của phương trình đều có dạng \(x^2 - 3x + 2\), nên ta có thể tách riêng phần chứa căn thức để giải: \(x^2 - 3x + 2 - (x^2 - 3x + 2) + \sqrt{x - 1} = 0\) \(\sqrt{x - 1} = 0\) Bây giờ, ta tìm giá trị của \(x\) để phương trình trên được thoả mãn: \(\sqrt{x - 1} = 0 \rightarrow x - 1 = 0 \rightarrow x = 1\) Kiểm tra điều kiện, ta thấy \(x = 1\) thoả mãn điều kiện xác định, nên đây là nghiệm của phương trình. Vậy phương trình có một nghiệm duy nhất là \(x = 1\).
Solving Equations with Newton's Method
The equation provided in the image is: \[ e^x + 7x + 11 = 20 \] To solve for \( x \), we’ll first simplify the equation by moving all terms to one side and setting the equation equal to zero: \[ e^x + 7x + 11 - 20 = 0 \] \[ e^x + 7x - 9 = 0 \] This equation isn't easily solvable with elementary algebraic methods because it mixes exponential and polynomial expressions. To solve for \( x \), we typically use numerical methods such as Newton's method, also known as the Newton-Raphson method. This method involves making an initial guess for the value of \( x \) and then iteratively improving that guess until a sufficiently accurate value is found. Here is how the Newton-Raphson method is applied in general: 1. Choose an initial guess \( x_0 \). 2. Calculate the next approximate value using the formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] where \( f(x) \) is the function for which we are trying to find a root, and \( f'(x) \) is its derivative. 3. Repeat step 2 using the new value of \( x \) until the difference between subsequent values of \( x \) is within a desired level of accuracy. The function \( f(x) \) based on the equation is: \[ f(x) = e^x + 7x - 9 \] The derivative of \( f(x) \) is: \[ f'(x) = e^x + 7 \] Without a computer or calculator, it's not feasible to carry out the Newton-Raphson method to a high level of accuracy, but in a general sense, we would begin with a reasonable guess for \( x \) and apply the method until the successive iterations converge on a solution. To properly solve for \( x \), this process would typically be performed using a calculator with capabilities of performing exponential and natural logarithm calculations, or a computer algorithm programmed to apply the Newton-Raphson method.
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