Problem 61 - 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.) $$ 9 x^{2}-30 x+c=0 $$

Step-by-Step Solution

Answer

The value of \(c\) is 25.

Step 1: Identify coefficients

For the quadratic equation in the form \[ax^2 + bx + c = 0,\] identify the coefficients: \(a = 9\), \(b = -30\), and \(c\) (unknown).

Step 2: Understand the discriminant

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 equal to 0.

Step 3: Set up the discriminant equation

Set the discriminant \(D\) equal to 0 for our given equation \(9x^2 - 30x + c = 0\): \[(-30)^2 - 4(9)c = 0.\]

Step 4: Solve for the discriminant

Simplify the discriminant equation: \[900 - 36c = 0.\]

Step 5: Solve for \(c\)

Rearrange the equation to isolate \(c\): \[36c = 900.\] Then divide both sides by 36 to find \(c\): \[c = \frac{900}{36} = 25.\]

Key Concepts

quadratic solutions

The solutions to a quadratic equation are the values of the variable that satisfy the equation. These values, also known as roots, are found by solving the equation set to zero in the form
ax^2 + bx + c = 0.
Quadratic equations generally have two solutions, but there are special cases when they can have one or no solutions at all. The nature of the solutions is dictated by the discriminant, which we will discuss next.

discriminant

The discriminant is a key part of the quadratic formula and plays a crucial role in determining the nature of the roots of a quadratic equation. The discriminant, denoted as D,
is found using the formula:
D = b^2 - 4ac.
Here's how the discriminant affects the solutions of a quadratic equation:

• If D > 0, the quadratic equation has two distinct real solutions.
• If D = 0, the quadratic equation has exactly one real solution (also known as a double root).
• If D < 0, the quadratic equation has two complex solutions.

Understanding the discriminant helps in predicting the number and type of solutions without actually solving the equation fully.

one rational solution

In certain cases, a quadratic equation will have exactly one rational solution. This special case occurs when the discriminant equals zero (D = 0). When D is zero, the quadratic equation has a perfect square, resulting in a single root. This root is rational if the coefficients a, b, and c
in the quadratic equation are rational numbers themselves.
For the example given, 9x^2 - 30x + c = 0, the hint guides us to set the discriminant to zero to find the value of c. Using the formula D = b^2 - 4ac, then setting it to zero and solving for c, verifies it. The steps show that when D = 0,
we get c = 25, resulting in exactly one rational solution for the quadratic equation.

coefficients in quadratic equations

Understanding the role of coefficients in quadratic equations is vital to both solving and interpreting these equations. A general quadratic equation is of the form ax^2 + bx + c = 0,
where a, b, and c
are coefficients.

• a: The coefficient of x^2, also known as the leading coefficient. Its value affects the