Problem 72 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The equation has 2 real solutions as the discriminant is positive.

Step 1: Identify the Coefficients

For the quadratic equation \(x^2 - rx + s = 0\), identify the coefficients: \(a = 1\), \(b = -r\), and \(c = s\).

Step 2: Write the Discriminant Formula

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\).

Step 3: Substitute Coefficients into Discriminant

Substitute the identified coefficients into the discriminant formula: \(D = (-r)^2 - 4(1)(s) = r^2 - 4s\).

Step 4: Analyze Conditions for Real Solutions

For a quadratic equation, the number of real solutions is determined by: - 2 real solutions if \(D > 0\)- 1 real solution if \(D = 0\)- No real solution if \(D < 0\).

Step 5: Apply Given Condition and Determine Outcome

You are given \(r > 2\sqrt{s}\). Square both sides to get \(r^2 > 4s\). Therefore, \(D = r^2 - 4s > 0\), implying that the equation has 2 real solutions.

Key Concepts

Number of Real Solutions

Understanding the number of real solutions in a quadratic equation is crucial in solving these equations. The discriminant provides valuable information. It's part of the quadratic formula that helps determine the nature of the solutions without actually finding them. The discriminant, denoted as \( D \), is calculated as \( D = b^2 - 4ac \) for any quadratic equation in the form \( ax^2 + bx + c = 0 \).

Here's a simple way to determine the number of real solutions:

• Two real and distinct solutions if \( D > 0 \)
• One real solution if \( D = 0 \)
• No real solutions if \( D < 0 \)

In the context of the equation \( x^2 - rx + s = 0 \) with the condition \( r > 2 \sqrt{s} \), we substitute to find \( D = r^2 - 4s \). Given \( r^2 > 4s \), we confirm \( D > 0 \), indicating two real solutions.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula allows us to find the roots or solutions of the equation \( ax^2 + bx + c = 0 \) by considering the coefficients \( a \), \( b \), and \( c \).

The quadratic formula includes the discriminant \( \sqrt{b^2 - 4ac} \) within its structure. This is where the number of real solutions is determined:

• The term \( \pm \sqrt{b^2 - 4ac} \) shows how solutions differ depending on the sign of the discriminant.
• For \( D > 0 \), this results in two roots because the square root yields a positive and negative value.
• For \( D = 0 \), the square root is zero, giving one root.
• For \( D < 0 \), no real number exists under the square root, indicating no real solutions.

Overall, the quadratic formula efficiently finds solutions while the discriminant within signifies the solution's nature.

Coefficients in Quadratic Equations

Coefficients in quadratic equations are vital in both forming these equations and finding their solutions. In the standard form \( ax^2 + bx + c = 0 \), the coefficients are:

• \( a \): The coefficient of \( x^2 \)
• \( b \): The coefficient of \( x \)
• \( c \): The constant term

These coefficients influence the overall shape and position of the graph of a quadratic equation.

Alongside providing structure, coefficients play a critical role in calculating the discriminant:\( D = b^2 - 4ac \). They determine the value of the discriminant and thus, the nature of the equation's solutions. In the context of \( x^2 - rx + s = 0 \):

• \( a = 1 \)
• \( b = -r \)
• \( c = s \)

Their values guide us to compute \( D = r^2 - 4s \). Therefore, understanding coefficients goes hand in hand with using the quadratic formula to solve realizing robust quadratic equation analysis.