Problem 59 - 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.) $$ a m^{2}+8 m+1=0 $$

Step-by-Step Solution

Answer

The value of \(a\) is 16.

Step 1: Identify the quadratic coefficients

For the quadratic equation \(a m^{2}+8 m+1=0\), identify the coefficients: \(a\) (coefficient of \(m^{2}\)), \(b = 8\) (coefficient of \(m\)), and \(c = 1\) (constant term).

Step 2: Use the discriminant formula

The discriminant of a quadratic equation \(am^2 + bm + c = 0\) is given by \(b^2 - 4ac\). For the equation to have exactly one rational solution, the discriminant must equal zero: \(b^2 - 4ac = 0\).

Step 3: Substitute the given values

Substitute \(b = 8\) and \(c = 1\) into the discriminant formula: \[ 8^2 - 4ac = 0 \] which simplifies to \[ 64 - 4a(1) = 0 \]

Step 4: Solve for the unknown coefficient \(a\)

Set up the equation: \[ 64 - 4a = 0 \] To solve for \(a\), isolate \(a\): \[ 64 = 4a \] \[ a = \frac{64}{4} \] \[ a = 16 \]

Key Concepts

Quadratic Coefficients

To start solving quadratic equations, it's important to first recognize the quadratic coefficients.
In the equation provided, \(a m^2 + 8m + 1 = 0\), the coefficients are the numbers in front of each term.
These coefficients help us understand the shape and position of the parabola that the quadratic equation represents.
Here's how the coefficients map out:

• \(a\) is the coefficient of \(m^2\)
• \(b = 8\) is the coefficient of \(m\)
• \(c = 1\) is the constant term

By identifying these values, we can use them in the discriminant formula, a crucial step in understanding the nature of the equation's solutions.

Discriminant Formula

The discriminant formula is central to analyzing quadratic equations.
It helps us determine the number and type of solutions for the equation.
The discriminant \( \Delta \), for any quadratic equation \( am^2 + bm + c = 0 \), is given by the formula:
\( \Delta = b^2 - 4ac \)
This formula uses the coefficients \(a\), \(b\), and \(c\) to evaluate the nature of the roots.
Here’s how it works:

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution. This is known as the double root or the repeated root.
• If \( \Delta < 0 \), there are no real solutions. Instead, there are two complex solutions.

Using the provided values \( b = 8 \) and \( c = 1 \), we substitute them into the discriminant formula to find \( a \).

One Rational Solution

To determine when a quadratic equation has exactly one rational solution, the discriminant must equal zero.
This simplifies our problem, as we know we need to set the discriminant formula to zero:
\( b^2 - 4ac = 0 \)
By substituting the identified coefficients \( b = 8 \) and \( c = 1 \) into the formula, we get:
\( 8^2 - 4a(1) = 0 \)
Simplifying further, we solve for \( a \):
\[ 64 - 4a = 0 \]
\[ 64 = 4a \]
\[ a = \frac{64}{4} \]
\[ a = 16 \]
Therefore, when \( a = 16 \), the quadratic equation \( 16m^2 + 8m + 1 = 0 \) will have exactly one rational solution.
This means the vertex of the equation touches the x-axis at just one point.