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. \(-4 x+c=-a x^2 ;\) two imaginary solutions

Short answer

An example pair of integer values for \(a\) and \(c\) that makes the quadratic equation \(-ax^2-4x+c=0\) have imaginary solutions is \(a = 1\) and \(c = 5\), yielding the equation \(-x^2 - 4x + 5 = 0\).

Step-by-step solution

Write down the general form of a quadratic equation

A quadratic equation has the general form \(ax^2+bx+c=0\). In the given equation \(-ax^2-4x+c=0\), \(a\) and \(c\) need to be determined and it's assumed that \(b = -4\).

Utilize the discriminant theory

For a quadratic equation to have imaginary roots, its discriminant \(b^2-4ac\) must be less than zero. Here, \(b = -4\) and we are solving for \(a\) and \(c\) such that \(b^2-4ac < 0\). This gives us \((-4)^2 - 4ac < 0\) or \(16 - 4ac < 0\).

Solve the inequality

Solving the inequality \(16 - 4ac < 0\) for two elements \(a\) and \(c\) in the set of integers can be achieved by trial and error. For instance, if \(a = 1\) and \(c = 5\), the inequality becomes \(16 - 4*1*5 = -4\), which is less than zero, thus satisfying the condition that the discriminant should be less than zero.

Write down the equation

The final quadratic equation can now be written using the values for \(a\) and \(c\) found. Thus, the equation becomes \(-x^2 -4x +5 = 0\).

Key concepts

Imaginary Solutions

When solving quadratic equations, you may encounter solutions that are not real numbers. These are known as **imaginary solutions**. Imaginary solutions arise when the equation under the square root in the quadratic formula is negative. This happens because square roots of negative numbers are not defined within real numbers, and require the use of the imaginary unit, denoted as **i**, where \(i^2 = -1\).
For example, solving the equation \(x^2 + 1 = 0\) involves taking the square root of \(-1\), leading to solutions like \(x = i\) and \(x = -i\). Imaginary solutions show up in pairs and indicate that the graph of the quadratic equation does not intersect the x-axis.

Discriminant

The **discriminant** is a crucial part of understanding the nature of solutions to quadratic equations. It is represented by the expression \(b^2 - 4ac\) in the quadratic formula. The discriminant helps in predicting whether the solutions are real or imaginary.

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

In our exercise, we needed two imaginary solutions, which means ensuring \(b^2 - 4ac < 0\). By setting \(b = -4\) and choosing appropriate integer values for \(a\) and \(c\), we found that \(a = 1\) and \(c = 5\) satisfy the condition \(16 - 4ac < 0\). This confirms that the quadratic equation has imaginary roots.

Quadratic Formula

The **quadratic formula** is a powerful tool for solving quadratic equations and can be written as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula gives the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\).
To use the quadratic formula, identify \(a\), \(b\), and \(c\) from the equation and substitute them into the formula. The term under the square root, \(b^2 - 4ac\), is the discriminant we mentioned earlier. Depending on its value, the solutions can be real or imaginary.
In the given quadratic equation \(-x^2 -4x +5 = 0\), we can identify \(a = -1\), \(b = -4\), and \(c = 5\) and then use the quadratic formula to find the solutions. Just remember, since the discriminant is negative, these solutions will indeed be imaginary.

Integer Values

Choosing appropriate **integer values** for the coefficients \(a\) and \(c\) is an essential step in forming a quadratic equation with specific solution characteristics.
In our exercise, the challenge was to ensure the quadratic equation produced imaginary solutions. To achieve this, we needed values of \(a\) and \(c\) that would make the discriminant \(b^2 - 4ac\) negative. By choosing \(a = 1\) and \(c = 5\), the inequality \(16 - 4ac < 0\) was satisfied, resulting in a negative discriminant.

• Choosing integer values often involves some trial and error, especially to ensure any specific requirements for the equation's solutions.
• In practice, this process strengthens the understanding of how different coefficients influence the nature of quadratic equations.

Therefore, understanding how to manipulate integer values for \(a\) and \(c\) is key to crafting equations with desired properties.