Problem 57 - 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.) $$ p^{2}+b p+25=0 $$

Step-by-Step Solution

Answer

The values of b are 10 or -10.

Step 1: Identify the Discriminant Formula

The discriminant for a quadratic equation in the form of \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). This value determines the nature of the roots of the equation.

Step 2: Set the Discriminant to Zero

According to the hint, for the quadratic equation to have exactly one rational solution, the discriminant should be zero. Therefore, set \(b^2 - 4ac = 0\).

Step 3: Substitute the Given Values

For the equation \(p^2 + bp + 25 = 0\), we have \(a = 1\), \(b = b\), and \(c = 25\). Substitute these values into the discriminant equation: \(b^2 - 4(1)(25) = 0\).

Step 4: Solve for b

Simplify the equation: \(b^2 - 100 = 0\). Add 100 to both sides to get \(b^2 = 100\). Taking the square root of both sides, we find two possible values for \( b \): \(b = 10\) or \(b = -10\).

Key Concepts

Discriminant

To understand how to solve any quadratic equation, you first need to grasp the concept of the discriminant. The discriminant is a critical part of the quadratic formula, providing valuable information about the nature of the roots. For any quadratic equation given by the form \(ax^2 + bx + c = 0\), the discriminant is represented by \(b^2 - 4ac\).

The discriminant reveals several important details about the roots of the equation:

• If the discriminant is positive (greater than 0), the equation has two different real and rational roots.
• If the discriminant equals zero, the equation has exactly one real and rational root – also known as a repeated or double root.
• If the discriminant is negative (less than 0), the equation has two complex roots.

In our exercise, we want the equation to have exactly one rational solution, so we set the discriminant to zero. This ensures the outcome we seek.

Rational Solutions

Rational solutions are solutions to an equation that can be expressed as a fraction or an integer. In the context of quadratic equations, whether the solutions are rational or not can be determined by the discriminant.

Here's why:

• When the discriminant is a perfect square and the quadratic formula yields rational numbers, the solutions are rational.
• When the discriminant is zero, it simplifies the quadratic formula to give a single rational solution, as \( \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) simplifies to \((-\frac{b}{2a})\).

In our step-by-step solution, after setting the discriminant to zero and solving for \(b\), we find it equals \(\text{10 or -10}\), ensuring that the equation \(p^2 + bp + 25 = 0\) has exactly one rational solution.

Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula not only allows you to find the roots of the equation but also uses the discriminant, \((b^2 - 4ac)\), to provide insight into the nature of the roots.

In the provided exercise, using the quadratic formula:

• We first identify \(a=1\), \(b=b\), and \(c=25\text{ in the equation p^2 + b p + 25 = 0.}\)
• By setting the discriminant to zero \(b^2-4(1)(25)=0\), we simplify to \(b^2 = 100\), thus, \(\text{b can be 10 or -10}\).
• Using either of these \(b-values\) in the quadratic formula, we see that it yields exactly one rational solution.

While it is possible to factorize the quadratic equation directly to find roots, the quadratic formula is a consistent method. It works every time, no matter how complex the equation may be.