Question

Solve the equation by completing the square. $$x^{2}+x-1=0$$

Short answer

The solutions for the quadratic equation are \(x = -0.5 + \sqrt{1.25}\) and \(x = -0.5 - \sqrt{1.25}\).

Step-by-step solution

Rewrite quadratic equation

Firstly, we have to rewrite the equation in the form \(x^2+bx+c=0\), which is in this case \(x^2 + x - 1 = 0\).

Complete the square

To complete the square, we add and subtract \((b/2)^2\) inside the brackets. So, we rewrite the equation as: \((x + (1/2))^2 - (1/2)^2 - 1 = 0\). This simplifies to \((x+0.5)^2 -1.25 = 0\).

Solve for x

Finally, solving for x involves isolating x on one side of the equation. By moving 1.25 to the right side of the equation and taking square root on both sides, we get two solutions: \(x = -0.5 + \sqrt{1.25}\) and \(x = -0.5 - \sqrt{1.25}\).

Key concepts

Quadratic Equations

A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. The general form of a quadratic equation is given by ax^2 + bx + c = 0, where a, b, and c are constants, and the variable x represents an unknown. Quadratic equations are fundamental in algebra and are characterized by their parabolic graphs that either open upwards or downwards.

When solving quadratic equations, there are several methods that can be employed, including factoring, using the quadratic formula, or completing the square. Each method has its own advantages depending on the specific equation. Completing the square is particularly useful when the quadratic equation does not factor neatly or when an exact decimal or radical solution is required.

Solve for x

Solving for x means finding the value(s) of the variable that make the equation true. In the case of quadratic equations, there can be zero, one, or two real solutions depending on the discriminant b^2 - 4ac. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution, which is repeated; and if it is negative, there are no real solutions, but two complex solutions.

Solving a quadratic equation by completing the square involves manipulation of the equation until one side forms a perfect square trinomial. This trinomial, when factored, yields a binomial squared. Isolating the variable after creating the perfect square trinomial requires taking the square root of both sides, which leads to two possible solutions, considering both the negative and positive square roots.

Algebraic Methods

Algebraic methods encompass a variety of techniques used to manipulate and solve equations. Completing the square is one such method which highlights the versatility and systematic approach of algebra. It involves rearranging the terms of the quadratic equation to form a perfect square, hence the term 'completing the square'.

This technique is not only a method for solving quadratics but also for understanding the properties of quadratic functions, such as vertex and axis of symmetry. Additionally, it sets the foundation for more advanced mathematics, including the derivation of the quadratic formula. Algebraic methods like this are essential for making complex equations more manageable and providing insights into the nature of mathematical relationships.