completing the square
Completing the square is a technique used in algebra to rewrite a quadratic expression in the form of a perfect square trinomial
Completing the square is a technique used in algebra to rewrite a quadratic expression in the form of a perfect square trinomial. This process is helpful for solving quadratic equations, graphing parabolas, and simplifying expressions.
To complete the square, you follow these steps:
1. Begin with a quadratic expression in the form of ax^2 + bx + c, where a, b, and c are constants.
2. Divide the entire expression by the coefficient of x^2, which is a. This step allows us to rewrite the equation with a leading coefficient of 1. For example, if you have 3x^2 + 6x + 5, divide the entire expression by 3 to get x^2 + 2x + 5/3.
3. Take half of the coefficient of x (which is b/2) and square it. This gives you (b/2)^2. Add this term to both sides of the equation. For example, if you have x^2 + 2x + 5/3, (2/2)^2 = 1, so you add 1 to both sides to get x^2 + 2x + 1 + 5/3.
4. Rearrange the terms inside the parentheses as a perfect square trinomial. In the example, (x + 1)^2 + 5/3.
5. Simplify the equation as needed, either by factoring the perfect square trinomial further or leaving it in the completed square form.
Completing the square is useful in solving quadratic equations by factoring. Once you have completed the square, you can rewrite the quadratic equation in the form of (x + p)^2 = q, where p and q are constants. This helps in finding the values of x that satisfy the equation.
Additionally, completing the square is essential when graphing parabolas. By rewriting the quadratic function in the form of y = a(x – h)^2 + k, where (h, k) is the vertex of the parabola, you can easily identify the vertex and other key features of the parabola.
In summary, completing the square is a method used to rewrite a quadratic expression in the form of a perfect square trinomial, enabling easier solving of quadratic equations and graphing of parabolas.
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