Completing the square is a technique used to rewrite a quadratic equation in a form that allows you to easily find the vertex of the parabola or solve for the roots. The core idea is to manipulate the equation to create a perfect square trinomial, which can then be factored into a squared binomial. This technique is incredibly useful for solving quadratic equations, graphing parabolas, and simplifying expressions.
Let's say you have a quadratic expression like $ax^2 + bx + c$. To complete the square, you focus on the $x^2$ and $x$ terms. If $a = 1$, you take half of the coefficient of the $x$ term (which is $b$), square it $(\frac{b}{2})^2$, and then add and subtract this value within the expression. This allows you to rewrite the first three terms as a perfect square binomial. If $a \neq 1$, you will need to factor out $a$ from the $x^2$ and $x$ terms first.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Perfect Square Trinomial | A. The point where the parabola changes direction. |
| 2. Quadratic Equation | B. An expression of the form $ax^2 + bx + c = 0$. |
| 3. Coefficient | C. A trinomial that can be factored into $(ax + b)^2$ or $(ax - b)^2$. |
| 4. Vertex | D. The number multiplied by a variable in an algebraic term. |
| 5. Constant | E. A fixed value that doesn't change. |
Completing the square helps transform a quadratic equation into _________ form. To complete the square, take half of the coefficient of the $x$ term, which is represented by _______, then ________ it. This creates a ________ ________ trinomial, which can then be factored.
Explain, in your own words, why completing the square is a useful technique for solving quadratic equations, even when factoring is not possible. Provide an example to illustrate your explanation.
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