Question

Find a possible pair of integer values for \(a\) and \(c\) so that the quadratic equation has the given solution(s). Then write the equation. \(a x^2+4 x+c=0\); two imaginary solutions

Short answer

The pair of integer values \(a = 1\) and \(c = 5\) yields a quadratic equation \(x^2 + 4x + 5 = 0\) that has two imaginary solutions.

Step-by-step solution

Understand what characterizes imaginary roots in a quadratic equation

In a quadratic equation of the form \(ax^2 + bx + c = 0\), if the discriminant \(b^2 - 4ac < 0\), then the equation has two imaginary solutions.

Apply conditions to the given equation

Apply the condition from step 1 to the given equation. We have \(a x^2 + 4x + c = 0\). Compare to the standard form, we can tell that \(b=4\) . The condition becomes \(b^2 - 4ac < 0 \rightarrow 16 - 4ac < 0\).

Solve for possible integer values

To satisfy \(16 - 4ac < 0\), integer values can be chosen arbitrarily as long as they satisfy the condition. For example, taking \(a = 1, c = 5\), when substituted in \(16 - 4ac\), we get \(16 - 4(1)(5) = -4 < 0\). Therefore, \(a = 1\) and \(c = 5\) is a valid pair of integer values.

Key concepts

Understanding Quadratic Equations

Quadratic equations might sound complex, but they're quite simple at their core. They are mathematical expressions of the form \[ ax^2 + bx + c = 0 \] where:

• \( a, b, \) and \( c \) are constants (with \( a eq 0 \))
• \( x \) is the variable that we need to solve for

The primary goal of solving a quadratic equation is to find the values of \( x \) that make the equation true. These are known as the roots or solutions. Sometimes, these solutions are real numbers, and sometimes, they are not. Having imaginary solutions means that there are no real numbers that satisfy the equation. Instead, the solutions involve imaginary numbers — numbers that include the term \( i \) (where \( i \) is the square root of \(-1\)).Quadratic equations can have different types of solutions depending on the discriminant (which we'll explore next). But the essence of quadratic equations lies in their ability to represent parabolic curves when graphed, with their solutions representing where these curves intersect the x-axis.

The Role of the Discriminant

The discriminant is a key concept in understanding the nature of solutions for quadratic equations. Represented by \( b^2 - 4ac \) in the quadratic formula, the discriminant helps us determine the type of roots an equation will have.For a quadratic equation \( ax^2 + bx + c = 0 \), here's how the discriminant affects the solutions:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, also known as a repeated or double root.
• If \( b^2 - 4ac < 0 \), there are two complex (imaginary) roots.

When the discriminant is less than zero, it indicates that the parabola does not intersect the x-axis and thus has no real solutions. Instead, you get imaginary solutions which involve the imaginary unit \( i \). Imaginary solutions occur when it's impossible to solve the equation with real numbers alone, showcasing the beauty and necessity of complex numbers in mathematics.

Finding Integer Values in Quadratic Solutions

Selecting integer values for \( a \) and \( c \) in a quadratic equation is sometimes required in math exercises or theoretical work. The challenge lies in ensuring these integers still satisfy the condition laid out by the discriminant.To derive imaginary solutions, we specifically seek integer values of \( a \) and \( c \) that make the discriminant negative:For our equation \[ ax^2 + 4x + c = 0 \]we set up the condition:\[ b^2 - 4ac < 0 \]In our example, replacing \( b \) with 4 gives:\[ 16 - 4ac < 0 \]The goal is to find integer values for \( a \) and \( c \) that meet this condition. In one possible solution, choosing \( a = 1 \) and \( c = 5 \) works perfectly, as \( 16 - 4 \times 1 \times 5 = -4 < 0 \), thus supporting two imaginary solutions. When solving similar types of problems, you should feel open to trying different integer combinations,but always ensure they satisfy the inequality for creating imaginary roots.