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Questions and Answers
Solve the following quadratic equation using the square root method: (x + 2)^2 = 9.
Solve the following quadratic equation using the square root method: (x + 2)^2 = 9.
- x = 1 or x = -5 (correct)
- x = 1 or x = -7
- x = 7 or x = -1
- x = 5 or x = -1
Which of the following quadratic equations can be solved using the square root method?
Which of the following quadratic equations can be solved using the square root method?
- x^2 + 4x - 5 = 0
- 3x^2 + 5x + 2 = 0
- -x^2 + 3 = 0 (correct)
- 2x^2 + 7 = 0
When taking the square root of both sides of an equation while solving a quadratic using the square root method, why is it essential to consider both the positive and negative square roots?
When taking the square root of both sides of an equation while solving a quadratic using the square root method, why is it essential to consider both the positive and negative square roots?
- It ensures we don't miss any potential solutions for the quadratic equation. (correct)
- It ensures we only get one solution for the quadratic equation.
- It ensures that the equation is balanced and remains equal on both sides after taking the square root.
- It's a mathematical rule that applies to all equations, not just quadratics solved using the square root method.
What is the solution to the equation 4x^2 - 16 = 0 using the square root method?
What is the solution to the equation 4x^2 - 16 = 0 using the square root method?
Which of the following is a correct step in solving the equation (x - 5)^2 = 25 using the square root method?
Which of the following is a correct step in solving the equation (x - 5)^2 = 25 using the square root method?
Flashcards
Square Root Method
Square Root Method
A technique to solve quadratic equations of the form ax² + c = 0.
Isolate the squared term
Isolate the squared term
Moving all other terms away from x² to one side of the equation.
Taking the square root
Taking the square root
Extracting square roots from both sides of the equation, noting both positive and negative roots.
Imaginary numbers in quadratics
Imaginary numbers in quadratics
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Study Notes
Solving Quadratics using the Square Root Method
- The square root method is a straightforward technique for solving quadratic equations where the equation fits the form ax² + c = 0, where 'a' and 'c' are constants.
- The crucial step involves isolating the squared term (x²) on one side of the equation.
- Once isolated, take the square root of both sides of the equation.
- Remember that when taking the square root of both sides, you must consider both the positive and negative square roots. This is crucial as there will be two solutions.
- If a variable is not isolated (e.g., 2x² + 5 = 7), first isolate the x² term.
- Example 1: Solve x² = 25. Taking the square root of both sides, we get x = ±5. Two solutions: x = 5 and x = -5.
- Example 2: Solve (x-3)² = 16. Taking the square root of both sides gives x-3 = ±4. Solving for x gives two possible solutions: x = 7 or x = -1.
Additional points and complexities
- If the constant on the right side of the equation is negative (e.g., x² = -9), then taking the square root results in imaginary numbers, which indicate the equation has no real solution. Example: x² = -9 has no real solutions (x = ±3i).
- For equations with more complex squared expressions (e.g., (x+1)² = 12), remember to isolate the expression containing the variable (x+1) before taking the square root.
- Equations like x² + 10x+ 25 = 8 can be solved this way if they can be factored into a perfect square. This means the equation can be expressed as (x+a)² = b.
- This method works best when the equation can be easily manipulated to the form x² = ...
- Always remember to check your answers by substituting them back into the original equation to ensure they are valid solutions.
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