Question

Write an equation of the form \(x^2=d\) that has (a) two real solutions, (b) one real solution, and (c) no real solution.

Short answer

\((a)\) An example of a quadratic equation with two real solutions is \(x^2 = 4\) with solutions \(x=2\) and \(x=-2\).\n \((b)\) An example of a quadratic equation with one real solution is \(x^2 = 0\) which has the solution \(x = 0\). \n \((c)\) An example of a quadratic equation with no real solution is \(x^2 = -1\).

Step-by-step solution

Form Equation with Two Real Solutions

Since we need two real solutions, we choose \(d\) to be any positive real number (but not zero). This ensures the square root of \(d\) will be a real number. For example, the equation \(x^2=4\) will have the solutions \(x=2\) and \(x=-2\), i.e., two real solutions.

Form Equation with One Real Solution

To get a quadratic equation with exactly one real solution, we make \(d=0\). This is because the square root of zero is defined and equals zero. So, the quadratic equation takes the form \(x^2=0\), which has exactly one real solution \(x=0\).

Form Equation with No Real Solution

In order to have a quadratic equation with no real solutions, we make \(d\) less than zero i.e., a negative number. Since the square root of a negative number is not a real number, there will be no real solutions for the equation. For instance, let's write the equation \(x^2=-1\). There are no real solutions for this equation.

Key concepts

Real Solutions of Quadratic Equations

When we encounter quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \), a primary question to ask is: How many real solutions does it have? A helpful starting point is recognizing that a quadratic equation has real solutions if the value of \( d \) in an equation of the form \( x^2 = d \) is zero or positive.
It's essential to remember that the nature of the solutions depends on the value of \( d \). If \( d > 0 \), the equation yields two distinct real solutions because a positive number has both a positive and negative square root. For example, \( x^2 = 4 \) translates to the solutions \( x = 2 \) and \( x = -2 \). When \( d = 0 \), however, the equation will always simplify to \( x = 0 \), giving us a single real solution. This is due to the fact that zero has exactly one square root, itself.
Understanding this principle is a crucial aspect of working with quadratic equations, as it sets the stage for determining the nature of the roots based on the discriminant, the part of the quadratic formula underneath the square root sign.

Zero-Product Property

A cornerstone of algebra, the zero-product property stipulates that if the product of two factors is zero, then at least one of the factors must be zero. This property is particularly useful when solving quadratic equations that can be factored. For example, consider \( (x - 2)(x + 2) = 0 \). According to the zero-product property, for this equation to hold true, either \( x - 2 = 0 \) or \( x + 2 = 0 \), yielding the solutions \( x = 2 \) or \( x = -2 \).
In other words, if a product equals zero, we can set each factor to zero and solve for the unknown to find the solutions. This property is frequently applied after factoring a quadratic equation into the form \( (x - p)(x - q) = 0 \). While this method is incredibly effective for finding real solutions, it's also a gateway to understanding scenarios where solutions might not be real.

Complex Numbers

But what happens when we encounter a quadratic equation where \( d < 0 \), such as \( x^2 = -1 \)? Since the square root of a negative number is not real, we enter the realm of complex numbers. Complex numbers include a real part and an imaginary part. The imaginary unit, denoted by \( i \), is defined by the property that \( i^2 = -1 \).
So, the equation \( x^2 = -1 \) would have the solutions \( x = i \) and \( x = -i \), which are complex numbers. Essentially, when a quadratic equation has no real solutions, it means that its solutions will be complex. To provide an example, \( x^2 = -4 \) has the solutions \( x = 2i \) and \( x = -2i \). Complex numbers are a profound expansion of the number system, ensuring that every quadratic equation has solutions, even if they aren't real. Understanding complex numbers is pivotal for not only solving specific equations but also for exploring advanced mathematical concepts and various applications within science and engineering.