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Magazine Chapter 9: Problem 107
Solve by completing the square. $$2 p^{2}+7 p=14$$
Short Answer
Expert verified
p = -\frac{7}{4} \text{卤} \frac{鈭�161}{4}
Step by step solution
01
Move the constant term
To start, move the constant term to the other side by subtracting 14 from both sides: \[2p^2 + 7p - 14 = 0\] becomes \[2p^2 + 7p = 14\].
02
Divide by the coefficient of the quadratic term
Divide all terms by 2, the coefficient of the quadratic term, to simplify the equation: \[p^2 + \frac{7}{2}p = 7\].
03
Complete the square
To complete the square, take half of the coefficient of p (which is \( \frac{7}{2} \)), square it, and add it to both sides of the equation: \(\frac{7}{2} \times \frac{1}{2} = \frac{7}{4} \)\(\frac{7}{4}^2 = \frac{49}{16} \)Adding \( \frac{49}{16} \) to both sides gives us: \[p^2 + \frac{7}{2}p + \frac{49}{16} = 7 + \frac{49}{16}\].
04
Simplify the right side
Combine the terms on the right side: \[7 = \frac{112}{16}\]So the equation becomes: \[p^2 + \frac{7}{2}p + \frac{49}{16} = \frac{112}{16} + \frac{49}{16}\]\[p^2 + \frac{7}{2}p + \frac{49}{16} = \frac{161}{16}\].
05
Write the perfect square trinomial as a square
Rewrite the left side as a perfect square trinomial: \( \text{(p + } \frac{7}{4}\text{)}^2 = \frac{161}{16} \).
06
Solve for p
Take the square root of both sides: \(\text{p + } \frac{7}{4}\text{ = } \frac{ \text{卤} \text{鈭�161}}{4} \)Then, isolate p by subtracting \(\frac{7}{4}\) from both sides: \[ p = -\frac{7}{4} \text{卤} \frac{ \text{鈭�161}}{4} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
A quadratic equation is any equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These equations generally have up to two real solutions. They can be graphically represented as a parabola. The solutions, also known as roots, are the x-values where the parabola intersects the x-axis. Quadratic equations appear often in various fields such as physics, engineering, and finance. To solve them, you can use methods like:
- Factoring
- Using the quadratic formula
- Completing the square
Perfect Square Trinomial
A perfect square trinomial is an expression that can be written as the square of a binomial. It looks like \((ax + b)^2\). For example, \(x^2 + 6x + 9\) is a perfect square trinomial, because it can be rewritten as \((x + 3)^2\). Identifying and forming perfect square trinomials are crucial steps in solving quadratic equations by completing the square. The steps involve:
The steps involve:
The steps involve:
- Taking half of the coefficient of the linear term (the term with x),
- Squaring that value,
- Adding it to both sides of the equation
Solving Quadratic Equations by Completing the Square
Completing the square allows us to solve quadratic equations by transforming them into a form that's easier to work with. Here's a recap of the steps from the exercise:
1. Move the constant term: \(2p^2 + 7p = 14\)
3. Complete the square:
4. Simplify the right side:
5. Write as a perfect square trinomial:
6. Solve for \(p\):
1. Move the constant term: \(2p^2 + 7p = 14\)
- We subtract the constant term \(14\) from both sides, focusing the equation on the quadratic and linear terms only.
- To simplify, we divide every term by the coefficient of \(p^2\), which is \(2\). This gives \(p^2 + \frac{7}{2}p = 7\).
3. Complete the square:
- Find half of the coefficient of \(p\), \(\frac{7}{4}\), and square it to get \(\frac{49}{16}\). Add it to both sides.
4. Simplify the right side:
- Combine and simplify any fractions. Here, \(7 + \frac{49}{16} = \frac{161}{16}\).
5. Write as a perfect square trinomial:
- Rewriting, we get \((p + \frac{7}{4})^2 = \frac{161}{16}\).
6. Solve for \(p\):
- Take the square root of both sides, considering both the positive and negative roots. Isolate \(p\), giving \(p = -\frac{7}{4} \text{卤} \frac{ \text{鈭�161}}{4}\).
