Completing the Square - Corbettmaths

Recent questions

• What is completing the square?

  Completing the square is a mathematical technique used to transform a quadratic equation into a specific format that makes it easier to analyze and solve. This method involves rewriting the equation in the form \((x + a)^2 + b\), where \(a\) and \(b\) are constants. By doing so, one can easily identify the vertex of the parabola represented by the quadratic function, which is crucial for determining its maximum or minimum values. This technique is particularly useful in various applications, including graphing quadratic functions and solving quadratic equations.

• How do I complete the square?

  To complete the square for a quadratic expression, you start by identifying the coefficient of the linear term. For example, in the expression \(x^2 + 8x + 18\), you take the coefficient of \(x\) (which is 8), halve it to get 4, and then square it to obtain 16. You then rewrite the expression by subtracting this squared value and adjusting the constant term accordingly. This process results in the expression being rewritten as \((x + 4)^2 + 2\). This method can be applied to any quadratic expression, allowing for a systematic approach to rewriting and solving these equations.

• Why is completing the square useful?

  Completing the square is useful for several reasons in mathematics. Firstly, it allows for the easy identification of the vertex of a quadratic function, which is essential for graphing the function accurately. Secondly, it simplifies the process of solving quadratic equations, as it transforms the equation into a form that can be easily manipulated to find the roots. Additionally, this technique is beneficial in calculus, particularly when dealing with optimization problems, as it helps in determining maximum and minimum values of quadratic functions. Overall, completing the square is a versatile tool in algebra and calculus.

• Can completing the square solve equations?

  Yes, completing the square can be used to solve quadratic equations effectively. By rewriting the equation in the form \((x - p)^2 = q\), where \(p\) and \(q\) are constants, one can isolate \(x\) and find its values. For instance, in the equation \(x^2 - 8x + 15 = 0\), completing the square allows you to rewrite it as \((x - 4)^2 - 1 = 0\). From there, you can solve for \(x\) by setting the squared term equal to 1, leading to the solutions \(x = 5\) and \(x = 3\). This method provides a clear pathway to finding the roots of quadratic equations.

• What are the steps to complete the square?

  The steps to complete the square involve a systematic approach to rewriting a quadratic expression. First, identify the quadratic and linear coefficients in the expression. Next, take half of the linear coefficient, square it, and add and subtract this value within the expression. This allows you to form a perfect square trinomial. Finally, simplify the expression by combining like terms. For example, in \(x^2 + 6x + 1\), you would halve 6 to get 3, square it to get 9, and rewrite the expression as \((x + 3)^2 - 8\). Following these steps ensures that you can complete the square for any quadratic expression efficiently.