Problem 67 - Algebra and Trigonometry

Question

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. \(x^{2}+r x-s=0 \quad(s>0)\)

Step-by-Step Solution

Answer

The equation has 2 real solutions.

Step 1: Identify the Coefficients

The quadratic equation given is \(x^2 + rx - s = 0\). Comparing this to the standard form \(ax^2 + bx + c = 0\), we identify the coefficients as \(a = 1\), \(b = r\), and \(c = -s\).

Step 2: Write the Formula for the Discriminant

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:\[\Delta = b^2 - 4ac\].

Step 3: Plug in the Coefficients into the Discriminant

Substitute \(a = 1\), \(b = r\), and \(c = -s\) into the discriminant formula:\[\Delta = r^2 - 4(1)(-s) = r^2 + 4s\].

Step 4: Analyze the Discriminant

For the quadratic equation's discriminant \(\Delta = r^2 + 4s\), note that since \(s > 0\), the term \(4s > 0\), ensuring that \(r^2 + 4s > 0\) for all values of \(r\).

Step 5: Conclude Number of Real Solutions

Because the discriminant \(\Delta = r^2 + 4s\) is always positive for \(s > 0\), the quadratic equation \(x^2 + rx - s = 0\) will always have 2 distinct real solutions.

Key Concepts

Number of Real Solutions

The discriminant is a crucial part of understanding the number of real solutions a quadratic equation might have. In a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is defined as \(\Delta = b^2 - 4ac\). This value helps us determine the nature of the roots.
For this specific problem, where \(s > 0\), we derived \(\Delta = r^2 + 4s\). Because the term \(4s\) is always positive, \(r^2 + 4s\) must also be positive.
When the discriminant \(\Delta > 0\), it indicates that the quadratic equation will have two distinct real solutions.

• Positive Discriminant: 2 distinct real solutions.
• Zero Discriminant: 1 real solution (roots are the same).
• Negative Discriminant: No real solutions (roots are complex).

In our case, the equation \(x^2 + rx - s = 0\) will always have two distinct real solutions when \(s\) is greater than zero.

Quadratic Formula

The quadratic formula is an essential tool in solving quadratic equations, a hallmark of algebra. Given a standard quadratic equation \(ax^2 + bx + c = 0\), the solutions for \(x\) can be found by using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula gives us the roots or solutions of the quadratic equation. The symbol \(\pm\) signifies that there are typically two solutions. One solution uses the addition, and the other uses subtraction.
The key component inside the square root, \(b^2 - 4ac\), is known as the discriminant. It directly influences the number and type of roots: real or complex.
Using the discriminant, we already determined that the equation \(x^2 + rx - s = 0\) with \(s > 0\) always yields two real solutions when applied to the quadratic formula.

Quadratic Coefficients

Coefficients in a quadratic equation are the constants \(a\), \(b\), and \(c\) found in its standard form: \(ax^2 + bx + c = 0\). These coefficients are critical as they affect all components of the quadratic equation, including the discriminant and the resulting solutions.
In the exercise \(x^2 + rx - s = 0\), we identify:

• \(a = 1\)
• \(b = r\)
• \(c = -s\)

Here, \(a\) is the coefficient of the \(x^2\) term, \(b\) is the coefficient of the \(x\) term, and \(c\) is the constant term. Proper identification of these coefficients allows for the correct application of the discriminant formula \(b^2 - 4ac\) and subsequently solving the quadratic equation with the quadratic formula.
This careful inspection of coefficients ensures that we apply the right values when calculating solutions and evaluating the nature of roots.