Problem 93 - Elementary Algebra

Question

Solve by completing the square. \(2 p^{2}+7 p=14\)

Step-by-Step Solution

Answer

The solutions are \(p = \frac{-7 + \sqrt{161}}{4}\) and \(p = \frac{-7 - \sqrt{161}}{4}\).

Step 1: Move Constant to the Other Side

First, subtract 14 from both sides to set the equation to zero: \(2p^2 + 7p - 14 = 0\).

Step 2: Divide by the Coefficient of the Quadratic Term

Divide every term by 2 to simplify the quadratic coefficient to 1: \(p^2 + \frac{7}{2}p - 7 = 0\).

Step 3: Move the Constant Term

Add 7 to both sides to isolate the quadratic and linear terms on the left side: \(p^2 + \frac{7}{2}p = 7\).

Step 4: Complete the Square

To complete the square, take half the coefficient of p, which is \(\frac{7}{4}\), and square it: \(\left(\frac{7}{4}\right)^2 = \frac{49}{16}\).Add and subtract \(\frac{49}{16}\) on the left side: \(p^2 + \frac{7}{2}p + \frac{49}{16} - \frac{49}{16} = 7\).Rewrite the left side as a perfect square: \((p + \frac{7}{4})^2 - \frac{49}{16} = 7\).

Step 5: Simplify

Add \(\frac{49}{16}\) to both sides to solve for the square term: \((p + \frac{7}{4})^2 = 7 + \frac{49}{16}\).Convert 7 to a fraction with a denominator of 16: \((p + \frac{7}{4})^2 = \frac{112}{16} + \frac{49}{16} = \frac{161}{16}\).

Step 6: Solve for p

Take the square root of both sides: \(p + \frac{7}{4} = \pm \sqrt{\frac{161}{16}}\).Simplify the right side: \(p + \frac{7}{4} = \pm \frac{\sqrt{161}}{4}\).Finally, subtract \(\frac{7}{4}\) from both sides: \(p = -\frac{7}{4} \pm \frac{\sqrt{161}}{4}\).

Key Concepts

quadratic equations

A quadratic equation is a second-order polynomial equation in a single variable. It has the form \[ax^2 + bx + c = 0\] where:\(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. In our exercise, the quadratic equation is given as \[2p^2 + 7p - 14 = 0\].Quadratic equations can have two solutions, one solution, or no real solutions at all. These solutions can be found using various methods, including:

• Factoring
• Using the quadratic formula
• Completing the square

algebraic methods

Algebraic methods are techniques used to manipulate equations and solve for unknown variables. These methods include simplification, factoring, expanding, and substitution. In this exercise, we use a specific algebraic method called 'completing the square' to solve the quadratic equation. Completing the square involves:

• Rearranging the terms of the equation
• Creating a perfect square trinomial
• Solving for the variable by isolating the square term

This method helps to transform the quadratic equation into a form that can be easily solved.

solving equations step by step

Solving equations step by step ensures that every part of the process is clear and easy to follow. Here's how we solved the given quadratic equation step-by-step:
1. We first moved the constant term to the other side: \[2p^2 + 7p = 14\].
2. Then, we divided every term by 2 to simplify the quadratic coefficient to 1: \[p^2 + \frac{7}{2}p - 7 = 0\].
3. We isolated the quadratic and linear terms: \[p^2 + \frac{7}{2}p = 7\].
4. We completed the square by adding and subtracting \[\frac{49}{16}\] to both sides: \[p^2 + \frac{7}{2}p + \frac{49}{16} - \frac{49}{16} = 7\].
5. Simplified the equation into a perfect square trinomial: \[(p + \frac{7}{4})^2 = \frac{161}{16}\].
6. Solved for \(p\) by taking the square root on both sides: \[p = -\frac{7}{4} \underline{\phantom{xxx}} \begin{cases}\text{+} \frac{\frac{\text{161}}{16}}{4} \text{-} \frac{\frac{\text{161}}{16}}{4}\text{ Then we simplified further}\text{.}\]

perfect square trinomials

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. In other words, it is an expression of the form \[(x + a)^2 = x^2 + 2ax + a^2\].To form a perfect square trinomial, we take half the coefficient of the linear term, square it, and add and subtract this square within the equation. For our equation \[p^2 + \frac{7}{2}p\], half the coefficient of \(p\) is \[\frac{7}{4}\]. When we square it, we get \[(\frac{7}{4})^2 = \frac{49}{16}\]. By adding and subtracting \[\frac{49}{16}\], we successfully formed the perfect square trinomial. This trinomial simplifies our quadratic equation and helps us solve it easily.