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Chapter 9: Problem 34

Solve each equation by completing the square. Give (a) exact solutions and (b) solutions rounded to the nearest thousandth. $$4 p+3=2 p^{2}+2 p$$

Short Answer

Expert verified
Exact solutions: \(p = \frac{1}{2} \text{卤} \frac{\text{鈭�7}}{2}\); Approximations: 1.822 and -0.822.

Step by step solution

01

Move all terms to one side

Rearrange the equation to get all terms on one side. Start with: \(4p + 3 = 2p^2 + 2p\)Subtract \(4p + 3\) from both sides to get: \(0 = 2p^2 - 2p - 3\)
02

Divide by the coefficient of the quadratic term

Divide each term by 2 to simplify the quadratic equation:\[p^2 - p - \frac{3}{2} = 0\]
03

Move the constant term

Move the constant term to the other side of the equation:\[p^2 - p = \frac{3}{2}\]
04

Complete the square

To complete the square, take half the coefficient of the linear term \(-1\), square it, and add it to both sides of the equation:Half of \(-1\) is \(-\frac{1}{2}\), and its square is \(\frac{1}{4}\). This gives:\[p^2 - p + \frac{1}{4} = \frac{3}{2} + \frac{1}{4} = \frac{7}{4}\]
05

Write left side as a square

Express the left side as a perfect square trinomial:\[(p - \frac{1}{2})^2 = \frac{7}{4}\]
06

Solve for p

Take the square root of both sides to solve for \(p\):\[p - \frac{1}{2} = \frac{\text{卤}\text{鈭�7}}{2}\]So, the solutions are:\[p = \frac{1}{2} + \frac{\text{鈭�7}}{2}\]\[p = \frac{1}{2} - \frac{\text{鈭�7}}{2}\]
07

Approximate the solutions

Calculate the decimal approximations of the solutions:\[p = \frac{1}{2} + \frac{\text{鈭�7}}{2} 鈮� 1.822\]\[p = \frac{1}{2} - \frac{\text{鈭�7}}{2} 鈮� -0.822\]

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Key Concepts

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

Completing the Square
Completing the square is a method used to solve quadratic equations by transforming them into a perfect square trinomial. This makes it easier to solve. The process involves:
  • Rearrange the equation to have all terms on one side.
  • Divide by the leading coefficient if it's not 1.
  • Move the constant term to the other side of the equation.
  • Add the square of half the coefficient of the linear term to both sides.
  • Rewrite the quadratic expression as a binomial squared.
  • Solve for the variable by taking the square root of both sides.

Remember, the main aim is to form a perfect square trinomial. For example, the quadratic equation given in the problem becomes:
\[ p^2 - p + \frac{1}{4} = \frac{7}{4} \text{, which transforms to} (p - \frac{1}{2})^2 = \frac{7}{4}.\]

Exact Solutions
Finding exact solutions involves solving the transformed perfect square trinomial. You do this by isolating the variable. In the provided exercise, we have (p - 1/2)^2 = 7/4.

To solve for p, take the square root of both sides of the equation:
\[p - \frac{1}{2} = \frac{\text{卤}\text{鈭�7}}{2}\]
Then, solve for p by adding 1/2 to both sides:
\

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Most popular questions from this chapter

Solve equation by using the square root property. Simplify all radicals. \((p-5)^{2}=40\)

Solve each equation by completing the square. Give (a) exact solutions and (b) solutions rounded to the nearest thousandth. $$3 r^{2}-2=6 r+3$$

Write each quotient in standard form. $$ \frac{i}{4-3 i} $$

Solve each quadratic equation for complex solutions by the square root property, with \(k<0 .\) Write solutions in standard form. $$ (x+6)^{2}=-7 $$

Solve each quadratic equation for complex solutions by the square root property, with \(k<0 .\) Write solutions in standard form. $$ (x-3)^{2}=-5 $$
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