Math Problem Statement
how to complete the square of a quadratic equation
Solution
To complete the square of a quadratic equation, you're transforming a quadratic expression into a perfect square trinomial, which can be factored easily. Here's the step-by-step process for completing the square for a quadratic equation of the form:
Steps:
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Move the constant to the other side: If the equation has a constant, move it to the other side by subtracting or adding it: If , first divide the entire equation by to make the coefficient of equal to 1:
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Add a value to both sides to complete the square: Take the coefficient of (which is ), divide it by 2, and square it. This will be the number you add to both sides of the equation: Add this value to both sides of the equation:
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Rewrite the left side as a perfect square: The left side is now a perfect square trinomial, so you can factor it as:
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Solve for : Take the square root of both sides: Finally, isolate :
Example:
Complete the square for :
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Move the constant to the other side:
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Take half of 6, square it, and add to both sides: Add 9 to both sides:
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Factor the left side:
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Solve for : So, , giving solutions:
Would you like more details or have any questions?
Here are 5 related questions to explore further:
- How is completing the square related to the quadratic formula?
- Can all quadratic equations be solved by completing the square?
- What are the advantages of completing the square over factoring?
- How does completing the square help in graphing quadratic functions?
- How does completing the square apply to real-world problems?
Tip: When solving quadratics, completing the square is especially useful when the quadratic equation cannot be easily factored.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Completing the Square
Formulas
Quadratic equation ax^2 + bx + c = 0
Completing the square formula: (x + b/2a)^2 = c
Theorems
Completing the Square Method
Quadratic Formula
Suitable Grade Level
Grades 9-12