Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Solve each quadratic equation by completing the square. $$x^{2}-2 x=2$$

Step-by-Step Solution

Answer

The solutions are \(x = 1 + i\) and \(x = 1 - i\).

Step 1: Move Constant to the Right

Start with the equation: \(x^2 - 2x = 2\). Move the constant term to the other side of the equation to get: \(x^2 - 2x = -2\).

Step 2: Identify Coefficient and Complete the Square

Look at the coefficient of the \(x\) term, which is \(-2\). Divide it by 2 and square it: \(\left(\frac{-2}{2}\right)^2 = 1\). Add this square to both sides of the equation: \(x^2 - 2x + 1 = -2 + 1\).

Step 3: Rewrite as a Perfect Square

Now the left side is a perfect square trinomial: \((x-1)^2 = -1\).

Step 4: Solve for x

Take the square root of both sides. Remember to consider both the positive and negative square roots: \(x - 1 = \pm \sqrt{-1}\). Since the square root of \(-1\) is \(i\), we write: \(x - 1 = \pm i\).

Step 5: Solve for x

Finally, solve for \(x\) by adding 1 to both sides: \(x = 1 \pm i\). This gives us the two solutions: \(x = 1 + i\) and \(x = 1 - i\).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Quadratic equations are fundamental in algebra and are widely used to model various phenomena. The standard form makes it easy to identify the coefficients, which helps in solving the equation using methods like factoring, completing the square, or using the quadratic formula.

The roots or solutions of a quadratic equation are the values of \(x\) that satisfy the equation, making the quadratic expression equal to zero. In real-world applications, these roots can represent anything from the time it takes for an object to fall to a certain height to the dimensions needed to optimize space.

When completing the square or using any other method, solving quadratic equations often leads us to solutions that can either be real numbers or complex numbers, depending on the discriminant \(b^{2} - 4ac\). If it's negative, the quadratic equation will have complex solutions.

Complex Numbers

Complex numbers come into play frequently when solving quadratic equations, especially when the discriminant \(b^{2} - 4ac\) is less than zero. A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).

In the context of quadratic equations, complex numbers expand the set of solutions from just real numbers to include numbers that involve the imaginary unit \(i\). For example, in the equation solved earlier, we had \(x = 1 \pm i\), indicating two complex solutions: \(x = 1 + i\) and \(x = 1 - i\).

Understanding how to work with complex numbers is crucial because it allows us to solve quadratic equations that do not intersect the x-axis on a Cartesian plane. Handling complex solutions means that we acknowledge scenarios beyond the scope of real numbers while still obeying algebraic principles. This can be especially useful in fields such as engineering and physics, where such solutions often have practical interpretations.

Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic expression that can be expressed as the square of a binomial. This means that after simplification, it will look like \((ax + b)^2\). Identifying and using perfect square trinomials is a significant strategy when completing the square.

For instance, in the equation \(x^2 - 2x + 1 = 0\), the left side can be rewritten as \((x - 1)^2\). This is a perfect square trinomial because it matches the form \((x + a)^2 = x^2 + 2ax + a^2\), with \(a = -1\) in this case.

Completing the square involves restructuring a quadratic equation into this form, making it easier to solve for \(x\) since we can then apply the property that if \((u)^2 = v\), then \(u = \pm \sqrt{v}\). Turning expressions into perfect square trinomials reduces the complexity of finding roots, and it is a vital technique for solving quadratic equations.