Example Question - square root method
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Solving Quadratic Equation with Square Root
The equation shown in the image appears to be a quadratic equation written as: \[ \sqrt{x^2 - 2x - 5} = x \] To solve this problem, we will first need to square both sides of the equation to eliminate the square root. Remember that when you square both sides of an equation, you may introduce extraneous solutions, so you'll need to check your answers in the original equation later. \[ (\sqrt{x^2 - 2x - 5})^2 = x^2 \] \[ x^2 - 2x - 5 = x^2 \] After squaring both sides, we notice that the x^2 terms on both sides of the equation are identical and they cancel out. \[ -2x - 5 = 0 \] Now, add 2x to both sides to isolate the constant term: \[ -5 = 2x \] To solve for x, divide both sides by 2: \[ x = -5 / 2 \] \[ x = -2.5 \] So the potential solution is x = -2.5. However, because we squared the equation to remove the square root, we need to verify that this potential solution satisfies the original equation: \[ \sqrt{(-2.5)^2 - 2(-2.5) - 5} \overset{?}{=} -2.5 \] Calculate the expression under the square root: \[ \sqrt{6.25 + 5 - 5} \] \[ \sqrt{6.25} \] We find the square root of 6.25: \[ \sqrt{6.25} = 2.5 \] Note that the square root of a positive number is positive, so the left side yields 2.5, not -2.5. Therefore, the potential solution x = -2.5 does not satisfy the original equation. This means that there is no solution to the original equation because the square root of a real number cannot equal a negative number. Thus, the original equation has no real solutions. If we needed to find complex solutions, we could continue the process knowing that the square root function would involve imaginary numbers when handling negative quantities, but based on the equation given, it seems we're dealing with real numbers only.
Solving Quadratic Equation with Percentage
Phép tính được cung cấp trong hình ảnh là: \[ (x - \frac{1}{5})^2 \pm 1 = 3.5\% \] Ta cần tìm giá trị của x. Đầu tiên, hãy đổi 3.5% thành dạng số thập phân: \[ 3.5\% = 0.035 \] Bây giờ phương trình của chúng ta trở thành: \[ (x - \frac{1}{5})^2 \pm 1 = 0.035 \] Giải quyết từng phần của phương trình, chúng ta có hai tình huống: 1. Khi cộng với 1: \[ (x - \frac{1}{5})^2 + 1 = 0.035 \] \[ (x - \frac{1}{5})^2 = 0.035 - 1 \] \[ (x - \frac{1}{5})^2 = -0.965 \] Phương trình này không có giải pháp nào trong tập số thực vì bạn không thể có một số thực mà khi bình phương lại có giá trị âm. 2. Khi trừ đi 1: \[ (x - \frac{1}{5})^2 - 1 = 0.035 \] \[ (x - \frac{1}{5})^2 = 0.035 + 1 \] \[ (x - \frac{1}{5})^2 = 1.035 \] Áp dụng căn bậc hai cho cả hai vế của phương trình để tìm x: \[ x - \frac{1}{5} = \pm\sqrt{1.035} \] Tiếp tục tìm căn bậc hai của 1.035 (vì phép tính này có thể làm tròn, hãy đảm bảo sử dụng máy tính để có kết quả chính xác nhất). Khi đã có giá trị này, hãy thêm 1/5 vào nó để tìm giá trị của x ứng với mỗi dấu: \[ x = \frac{1}{5} \pm\sqrt{1.035} \] Hãy nhớ rằng bạn chỉ có thể tính toán giá trị x trong phạm vi các số thực; kết quả âm khi bình phương không thể xảy ra với các số thực.
Solving Absolute Value Equations
The equation provided in the image is |2x - 3|^2 = 16. To solve for x, we need to take the square root of both sides of the equation and then solve the resulting absolute value equation. Starting with the equation: |2x - 3|^2 = 16 Take the square root of both sides to eliminate the square on the left side: |2x - 3| = √16 Since the square root of 16 is 4, we have: |2x - 3| = 4 The absolute value equation |2x - 3| = 4 can split into two possible equations because the absolute value of a number can be both positive and negative: 2x - 3 = 4 or 2x - 3 = -4 For the first equation: 2x - 3 = 4 2x = 7 x = 7/2 x = 3.5 For the second equation: 2x - 3 = -4 2x = -1 x = -1/2 x = -0.5 Therefore, there are two possible values for x: 3.5 and -0.5.
Solving Quadratic Equation with Square Root
The equation provided in the image is: √x - x = 0 To solve for x, start by isolating the square root term: √x = x Now, square both sides of the equation to eliminate the square root: (√x)^2 = x^2 This simplifies to: x = x^2 Rearrange the equation to set it to zero and find the values of x that satisfy the equation: x^2 - x = 0 Factor x out of the expression: x(x - 1) = 0 Now, set each factor equal to zero and solve for x: x = 0 x - 1 = 0 => x = 1 The solutions are x = 0 and x = 1. However, you must check these solutions in the original equation because squaring both sides can introduce extraneous solutions. For x = 0: √0 - 0 = 0 => 0 = 0 (true) For x = 1: √1 - 1 = 0 => 1 - 1 = 0 (true) Both solutions are valid, so x can be 0 or 1.
Solving Quadratic Equation x^2 - 7 = 0
The equation given in the image is: x^2 - 7 = 0 To solve for x, follow these steps: 1. Add 7 to both sides of the equation to isolate the x^2 term: x^2 - 7 + 7 = 0 + 7 x^2 = 7 2. Take the square root of both sides to solve for x: √(x^2) = ±√7 x = ±√7 Therefore, the solutions for x are x = √7 and x = -√7.
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