Exam Questions – Completing the square

Exam Questions – Completing the square

Introduction to Completing the Square

• Completing the square is a technique used in solving quadratic equations, optimising functions, and evaluating integrals.
• It creates a perfect square trinomial from a quadratic equation.

Application in Quadratic Equations

• A quadratic equation of the form ax^2 + bx + c is rewritten as (x+h)^2 + k through completing the square.
• To complete the square of a quadratic equation, we look at the b coefficient primarily. Half the coefficient of x creates h. The square of h is then added and subtracted within the brackets to create a perfect square.

Examples

• For example, the quadratic equation x^2 + 6x + 5 can be written as (x + 3)^2 - 4. This is obtained by setting h to 3 (half of 6), and forming a perfect square using h.
• The expression (x + h)^2 forms a perfect square, while -4 adjusts for the constant term in the original expression.

Solving Quadratic Equations

• Once the quadratic equation is in the form of (x+h)^2 + k, solving for x becomes straightforward.
• Take the square root of both sides, remembering to consider both positive and negative roots, then rearrange the equation.

Examples

• If we take the equation (x + 3)^2 - 4 = 0, we add 4 to both sides to get (x + 3)^2 = 4. Solving for x gives us x=-3-2=-5, and x=-3+2=-1.

Conclusion

• The technique of completing the square transforms a quadratic equation into a form that makes it easier and more convenient to solve.
• It involves altering the equation into a perfect square, a technique that is indispensably useful in tackling quadratic equations in algebra and in various mathematical applications.