Problem 62 - Beginning and Intermediate Algebra

Question

Find the value of a, b, or c so that each equation will have exactly one rational solution. (Hint: The discriminant must equal 0 for an equation to have one rational solution.) $$ 4 m^{2}+12 m+c=0 $$

Step-by-Step Solution

Answer

The value of c is 9.

Step 1: Identify the coefficients

For the quadratic equation in the form \[ax^2 + bx + c = 0\], identify the coefficients. Here, the equation is \[4m^2 + 12m + c = 0\], so we have: a = 4, b = 12, and c is unknown.

Step 2: Use the discriminant formula

Recall that the discriminant of a quadratic equation \[ax^2 + bx + c = 0\] is given by: \[D = b^2 - 4ac\]. For the equation to have exactly one rational solution, the discriminant must be zero \(D = 0\).

Step 3: Set the discriminant to zero

Using the coefficients identified, substitute them into the discriminant formula: \[D = 12^2 - 4(4)(c) = 0\]. Simplify and solve for c.

Step 4: Solve for c

\[144 - 16c = 0\] Reorganize the equation: \[16c = 144\] Divide both sides by 16: \[c = 9\]

Key Concepts

Quadratic Equation

A quadratic equation is any equation that can be written in the form \[ax^2 + bx + c = 0\]. This type of equation features a variable, usually represented by \(x\), that is squared.
In the exercise given, the quadratic equation is \[4m^2 + 12m + c = 0\]. Here, the variable is \(m\) and the equation has three coefficients: \(a = 4\), \(b = 12\), and \(c\) which is unknown.

Quadratic equations can have 0, 1, or 2 solutions depending on the value of their discriminant.
This characteristic makes them a foundational concept in algebra and necessary for understanding more complex equations.

Rational Solution

A rational solution in the context of quadratic equations refers to a solution that can be expressed as a ratio of two integers, like \(3/4\) or \(-2\).
In simpler terms, it is a solution that is not a complex or infinite decimal.
For a quadratic equation to have exactly one rational solution, as the exercise hints, the discriminant \(D\) must be zero.
In the formula \[D = b^2 - 4ac\], when the discriminant equals zero, the quadratic equation \[ax^2 + bx + c = 0\] simplifies in a way that it touches the x-axis at one unique point.
This unique point is our rational solution.

Coefficients

Coefficients are the numerical or constant factors that multiply the variables in an equation.
In the quadratic equation \[4m^2 + 12m + c = 0\], we have three coefficients:

• \(a = 4\), which is the coefficient multiplying \(m^2\)
• \(b = 12\), which is the coefficient multiplying \(m\)
• \(c\) is the constant term that we need to find

Understanding these coefficients is crucial for setting up the discriminant formula \[D = b^2 - 4ac\].
In this exercise, by setting the discriminant equal to zero (\(D = 0\)), and solving the equation \[12^2 - 4(4)(c) = 0\], we found that \(c = 9\).