Question

Can you solve an equation by completing the square when the equation has two imaginary solutions? Explain.

Short answer

Yes, you can solve an equation by completing the square even when the equation has two imaginary solutions. It's because the method of completing the square ultimately leads to the quadratic formula, which accommodates for imaginary solutions.

Step-by-step solution

Understand Completing the Square

Completing the square is a method used to solve quadratic equations. It involves rearranging the equation in a form that allows you to directly identify the solutions of the equation, either real or complex. It's the same technique you use when transforming a quadratic equation into vertex form: \(y=a(x-h)^2+k\). Here, 'h' and 'k' represent the vertex of the parabola while 'a' defines whether it opens up or down and the steepness of the curve.

Understand Imaginary Solutions

Imaginary solutions occur in quadratic equations when the discriminant is negative. The discriminant is the value under the square root in the quadratic formula, which is \(b^2 - 4ac\). If this value is negative, then the solutions of the equation are complex, since squaring any real number can never produce a negative value.

Concluding Whether Completing the Square Works for Equations with Imaginary Solutions

Since completing the square can lead to the quadratic formula, which certainly accounts for the presence of complex solutions (due to negative discriminants), it's true to say that completing the square can be used to solve an equation with two complex solutions.

Key concepts

Imaginary Solutions

In mathematics, imaginary solutions arise from quadratic equations when the discriminant is negative. This might sound complicated, but let’s break it down. The discriminant in a quadratic equation is the part under the square root in the quadratic formula: \( b^2 - 4ac \). It basically tells you how many and what kind of roots or solutions your quadratic equation has. When this value is negative, we can't find real roots because you cannot take the square root of a negative number using real numbers. Instead, we introduce imaginary numbers. A common symbol for this is \( i \), where \( i \) represents the square root of \(-1\).

Once you understand imaginary numbers, solving equations with imaginary solutions becomes much easier. These solutions often appear as a pair of complex numbers, but more on that later. The key takeaway here is understanding the role of the discriminant when it’s negative, indicating the presence of imaginary solutions.

Quadratic Equations

Quadratic equations are a form of polynomial equations of the second degree, usually written as \( ax^2 + bx + c = 0 \). They are fundamental in algebra and have a range of applications from physics to finance. Solving these equations can give us the roots, or solutions, which are where the parabola, the graphical representation of a quadratic equation, crosses the x-axis.

There are various methods to solve quadratic equations: factoring, using the quadratic formula, and completing the square. The method of completing the square is useful because it rearranges the equation into a perfect square trinomial \( (x-h)^2 \), which then helps in easily finding the vertex form \( y = a(x-h)^2 + k \). Completing the square also sets the stage for understanding whether a quadratic will have real or imaginary solutions, by showing clearly the expression inside the square root in the quadratic formula. It is a powerful technique especially when dealing with complex or imaginary solutions.

Complex Solutions

Complex solutions in quadratic equations arise when solving equations that yield a negative discriminant under the square root in the quadratic formula \( b^2 - 4ac \). They consist of a real part and an imaginary part, and are expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, multiplied by \( i \), the imaginary unit.

When a quadratic equation results in complex solutions, it indicates the parabola doesn't cross the x-axis on a graph, meaning there are no real x-intercepts. Completing the square remains relevant here, enabling you to clearly see the shift needed to form the square trinomial even with a negative discriminant. Hence, completing the square helps to reveal the nature of the solutions by making explicit the complex form \( a \pm bi \). Solving the equation this way helps solidify the concept that each quadratic equation is solvable, just sometimes in the complex number plane rather than the real number line.