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Chapter 0: Problem 44

Solve. $$2 p^{2}-5 p=1$$

Short Answer

Expert verified
The solutions are \( p = \frac{5 + \sqrt{33}}{4} \) and \( p = \frac{5 - \sqrt{33}}{4} \).

Step by step solution

01

- Write Down the Equation

Start with the given quadratic equation: \[2p^{2} - 5p = 1\]
02

- Move All Terms to One Side

Subtract 1 from both sides of the equation to set it to zero: \[2p^{2} - 5p - 1 = 0\]
03

- Identify Coefficients

Identify the coefficients for the quadratic equation form \(ap^{2} + bp + c = 0\). Here, \(a = 2\), \(b = -5\), and \(c = -1\).
04

- Use the Quadratic Formula

The quadratic formula is given by \[p = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\]. Plug in the coefficients: \[a = 2, b = -5, c = -1\]
05

- Calculate the Discriminant

First, calculate the discriminant: \(b^{2} - 4ac\). \[(-5)^{2} - 4(2)(-1) = 25 + 8 = 33\]
06

- Solve for p

Use the quadratic formula components: \[p = \frac{-(-5) \pm \sqrt{33}}{2 \times 2} = \frac{5 \pm \sqrt{33}}{4}\]
07

- Write the Solutions

Therefore, the solutions to the equation are: \[p = \frac{5 + \sqrt{33}}{4}\] and \[p = \frac{5 - \sqrt{33}}{4}\]

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Key Concepts

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

quadratic formula
The quadratic formula helps you find the roots of any quadratic equation. A quadratic equation is typically written as \[ax^2 + bx + c = 0\], where \`a\`, \`b\`, and \`c\` are coefficients. To solve for \`p\`, plug the values of \`a\`, \`b\`, and \`c\` into the formula \[p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].Let's break it down:
    \t
  • \`b\` becomes negative. For instance, if \( b = -5 \), it turns into 5 inside the formula.\t
  • Calculate the discriminant, \( b^2 - 4ac \).\t
  • This discriminant is then placed inside the square root function.
The quadratic formula simplifies the process of finding roots, especially when factoring or other methods are complicated.
discriminant
The discriminant helps you determine the nature of the roots of a quadratic equation. It is computed as \( b^2 - 4ac \). Here are key points to remember:
    \t
  • If the discriminant is positive \( \Delta > 0 \), the equation has two distinct real roots.\t
  • If the discriminant is zero \( \Delta = 0 \), the equation has exactly one real root.\t
  • If the discriminant is negative \( \Delta < 0 \), the equation has two complex roots.
In our equation \( 2p^2 - 5p - 1 = 0 \), the discriminant is calculated as follows:\( (-5)^2 - 4(2)(-1) = 25 + 8 = 33 \)Since \( 33 \) is positive, the equation has two distinct real roots: \( \frac{5 + \sqrt{33}}{4} \) and \( \frac{5 - \sqrt{33}}{4} \).
coefficients
Coefficients are the numerical values that multiply the variables in a quadratic equation. In \[ 2p^2 - 5p - 1 = 0 \], the coefficients are:
    \t
  • \`a\` (coefficient of \`p^2\`) = 2\t
  • \`b\` (coefficient of \`p\`) = -5\t
  • \`c\` (constant term) = -1
Identifying these coefficients is crucial. They are substituted into the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots. By plugging in \( a = 2 \), \( b = -5 \), and \( c = -1 \), you can solve the quadratic equation step by step. Correct coefficients ensure accurate calculations of the roots or solutions for the equation.

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