Problem 60 - 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.) $$ a t^{2}+24 t+16=0 $$

Step-by-Step Solution

Answer

a = 9

Step 1: Identify the coefficients

The given quadratic equation is a t^{2} + 24 t + 16 = 0Identify the coefficients from this equation:a = a b = 24 c = 16

Step 2: Recall the condition for exactly one rational solution

For a quadratic equation to have exactly one rational solution, the discriminant must be equal to zero. The discriminant delta = b^{2} - 4ac must be zero.

Step 3: Calculate the discriminant

Substitute the values of a, b, and c into the discriminant formula:delta = 24^{2} - 4 * a * 16

Step 4: Set discriminant to zero

To have exactly one rational solution:24^{2} - 4 * a * 16 = 0

Step 5: Solve for the unknown coefficient a

Solve the equation to find the value of a:576 - 64a = 0Isolate a:64a = 576a = 9

Key Concepts

quadratic equation

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \). The general structure includes three main components: the squared term \( ax^2 \), the linear term \( bx \), and the constant term \( c \). Quadratic equations are fundamental in algebra. They can appear in various real-world scenarios.
To solve quadratic equations, you can use methods like factoring, completing the square, or the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The solutions to a quadratic equation are the values of \( x \) that make the equation true. These solutions can be real or complex numbers.
In our given problem, the quadratic equation is \( a t^2 + 24 t + 16 = 0 \). Here, we need to determine the coefficient \( a \) that makes the equation have one rational solution.

rational solution

A rational solution refers to a solution that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q eq 0 \). For a quadratic equation to have exactly one rational solution, the discriminant must be zero. The discriminant \( \Delta \) is part of the quadratic formula and is represented by \( \Delta = b^2 - 4ac \).
If \( \Delta = 0 \), the quadratic equation has exactly one solution, which is also rational. This is because the \( \sqrt{\Delta} \) term in the quadratic formula disappears, leaving us with \[ x = \frac{-b}{2a} \].
In the given problem, setting the discriminant equal to zero helps us find a specific value of \( a \) (i.e., \( a = 9 \)), ensuring that the equation \( a t^2 + 24 t + 16 = 0 \) has one rational solution.

coefficients

Coefficients are the numerical values multiplying the variables in an equation, setting their length or magnitude. In our quadratic equation \( a t^2 + 24 t + 16 = 0 \), the coefficients are \( a \), \( 24 \), and \( 16 \).
Each coefficient plays a crucial role.

• \( a \) is the coefficient of the squared term \( t^2 \).
• \( 24 \) is the coefficient of the linear term \( t \).
• \( 16 \) is the constant term.

Modifying these coefficients changes the shape and position of the parabola represented by the quadratic equation.
In the current problem, we used the discriminant to find the missing coefficient \( a \) so that the equation achieves exactly one rational solution. By solving \( 24^2 - 4a*16 = 0 \), we found \( a \) to be 9, confirming that every term's contribution results in a single solution.