uadratic equations are one of the fundamental concepts in algebra. They are often used to solve real-life problems involving quantities that vary as the square of another quantity. Completing the square is an important technique that helps in solving quadratic equations. In this article, we will delve into the details of completing the square, providing answers to common questions about this method.

What does it mean to complete the square?

Completing the square is a technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This process involves manipulating the given quadratic equation to obtain a quadratic expression that can be factored as a perfect square.

Why is completing the square used?

Completing the square is a powerful method for solving quadratic equations since it allows us to easily find the roots of the equation. By converting the quadratic equation into a perfect square trinomial, we can factor it and determine the values of x where the equation equals zero.

How is completing the square done?

To complete the square, follow these steps:
i. Start with a quadratic equation in the standard form: ax^2 + bx + c = 0.
ii. Divide the entire equation by a, if necessary, to make the coefficient of x^2 equal to 1.
iii. Move the constant term (c) to the other side of the equation.
iv. Add the square of half the coefficient of x to both sides of the equation.
v. Factor the resulting trinomial as a perfect square.
vi. Take the square root of both sides and solve for x.

Can you provide an example?

Certainly! Let’s take the quadratic equation x^2 + 6x = -3 as an example.

i. Since the coefficient of x^2 is already 1, we proceed to the next step.
ii. Move the constant term to the other side: x^2 + 6x + 3 = 0.
iii. Now we add the square of half the coefficient of x (3^2 = 9) to both sides: x^2 + 6x + 9 + 3 = 9.
iv. We then rearrange the equation: (x + 3)^2 = 9.
v. Factor the trinomial: (x + 3)^2 = 3^2.
vi. Take the square root of both sides: x + 3 = ±3.
vii. Finally, solve for x: x = -3 ± 3.

So, the roots of the quadratic equation x^2 + 6x + 3 = 0 are x = -6 and x = 0.

Are there any special cases when completing the square?

Yes, when the coefficient of x is even, an additional step is required. Let’s consider the quadratic equation 2x^2 + 8x + 4 = 0.

i. Divide the equation by the coefficient of x^2: 2(x^2 + 4x + 2) = 0.
ii. Move the constant term to the other side: x^2 + 4x = -2.
iii. Add the square of half the coefficient of x (2^2 = 4) to both sides: x^2 + 4x + 4 = 2.
iv. Rearrange and factor the trinomial: (x + 2)^2 = 2.
v. Take the square root and solve for x: x + 2 = ±√2.
vi. Solve for x: x = -2 ± √2.

Completing the square is a fundamental method in solving quadratic equations. It transforms a quadratic expression into a perfect square trinomial, making it easier to solve for the values of x. By following the steps outlined in this article, you can confidently complete the square and find the roots of given quadratic equations. With practice, this technique will become second nature, empowering you to tackle a wide range of quadratic equations effectively.

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