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Chapter 3: Problem 25

Find the value(s) of \(p\) for which the equation \(2 x^{2}+p x+1=0\) has one real solution.

Short Answer

Expert verified
The value(s) of \(p\) are \(p = 2\sqrt{2}\) and \(p = -2\sqrt{2}\).

Step by step solution

01

Understand the Condition for One Real Solution

For a quadratic equation of the form \(ax^2 + bx + c = 0\) to have only one real solution, the discriminant must be zero. The discriminant \(\Delta\) for the quadratic equation is given by \(b^2 - 4ac\). When \(\Delta = 0\), the quadratic equation has exactly one real solution.
02

Identify the Coefficients

In the equation \(2x^2 + px + 1 = 0\), identify the coefficients: \(a = 2\), \(b = p\), and \(c = 1\). These coefficients will be used in the formula for the discriminant.
03

Set the Discriminant to Zero

Using the formula \(b^2 - 4ac\), set the discriminant to zero: \[ p^2 - 4 \cdot 2 \cdot 1 = 0 \]This ensures the quadratic equation has one real solution.
04

Simplify the Equation

Simplify the equation from Step 3:\[ p^2 - 8 = 0 \]This simplifies to \(p^2 = 8\).
05

Solve for \(p\)

To find \(p\), take the square root of both sides of the equation \(p^2 = 8\): \[ p = \pm \sqrt{8} \]Simplify \(\sqrt{8}\) to get \(p = \pm 2\sqrt{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discriminant
The discriminant is a key component in determining the number and type of solutions for a quadratic equation. For any quadratic equation in the form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is calculated using the formula \( b^2 - 4ac \). This value tells us about the nature of the roots or solutions:
  • If \( \Delta > 0 \), the equation has two distinct real solutions.
  • If \( \Delta = 0 \), there is exactly one real solution, meaning the graph of the quadratic touches the x-axis at one point.
  • If \( \Delta < 0 \), no real solutions exist as the graph does not intersect the x-axis.
Therefore, setting the discriminant to zero is crucial when seeking solutions where the quadratic equation has a solitary real root. This understanding is used in our exercise to find specific values of \( p \) such that the equation \( 2x^2 + px + 1 = 0 \) has one real solution.
Real Solutions
Real solutions or real roots of a quadratic equation are the values of \( x \) where the quadratic equation evaluates to zero. These are the points where the graph of the quadratic function intersects or touches the x-axis.
  • When a quadratic equation has one real solution, it means the vertex of the parabola lies on the x-axis. In practical terms, it's the solution to the equation \( ax^2 + bx + c = 0 \) when the discriminant is zero.
  • The real solution is also referred to as a double root or a repeated root, because it effectively counts as two identical solutions in the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • In our specific example, finding \( p \) such that the equation has one real solution ensures that the parabola \( 2x^2 + px + 1 = 0 \) just touches the x-axis at its vertex.
Coefficients
Coefficients are the numbers that multiply the variables in the equation. For the quadratic equation \( ax^2 + bx + c = 0 \), \( a \), \( b \), and \( c \) are the coefficients linked to \( x^2 \), \( x \), and the constant term, respectively.
  • The coefficient \( a \) dictates the "width" and direction (upwards or downwards) of the parabola. In the exercise, \( a = 2 \), indicating a parabola that opens upwards and is relatively narrow.
  • \( b \) impacts the location of the vertex and the axis of symmetry of the parabola. Here, \( b = p \), making it our variable of interest to find when the equation has one real solution.
  • The constant \( c \) affects the y-intercept, which is the point where the graph crosses the y-axis. In our equation, \( c = 1 \), ensuring it does not shift vertically.
Recognizing these coefficients is instrumental not only in calculating the discriminant but also in understanding the graph's geometry and behavior.

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