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Chapter 5: Problem 23

Solve each equation. $$ 4 p^{2}-16=0 $$

Short Answer

Expert verified
The solutions are p = 2 or p = -2.

Step by step solution

01

- Isolate the quadratic term

Start by moving the constant term to the other side of the equation. Add 16 to both sides of the equation: 4p^2 - 16 + 16 = 0 + 16 which simplifies to: 4p^2 = 16.
02

- Solve for the squared term

To isolate the squared term, divide both sides by 4: \( \frac{4p^2}{4} = \frac{16}{4} \) which simplifies to: \( p^2 = 4 \).
03

- Find the roots by taking the square root

To solve for \( p \), take the square root of both sides of the equation: \( p = \text{卤} \right.\sqrt{4} \) This results in: \( p = 2 \) or \( p = -2 \).

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Key Concepts

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

quadratic term
A quadratic term is a term of an equation where the variable is raised to the power of two. These terms can appear anywhere in the equation and are characterized by their squared variable, such as in the term 4p^2 from the initial equation. Quadratic terms determine the parabolic shape of the equation's graph. In simpler terms, they tell you the direction and the width of the parabola. The quadratic term is crucial because it sets the foundation for solving quadratic equations.
square root
The square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because 2 * 2 = 4. When solving quadratic equations, taking the square root is a key step in isolating the variable. From our exercise, after isolating the squared term to get p^2 = 4, we take the square root of both sides of the equation. It's important to remember that square roots have both positive and negative roots, which gives us two solutions: positive and negative.
isolating variables
Isolating variables means rearranging the equation to express the variable of interest on one side of the equation. This generally involves moving constants and coefficients to the opposite side. In the given exercise, isolating the quadratic term is the first step: move the constant (which is -16) to the other side by adding it to both sides. We then have 4p^2 = 16. The next isolation step involves dividing both sides by 4 to isolate the squared term, resulting in p^2 = 4. This process simplifies the equation and makes it easier to solve.
roots of equations
The roots of equations are the solutions for the variable that satisfy the equation. For quadratic equations, roots are found after simplifying and isolating the squared term, and taking the square root of both sides. In the exercise, after simplifying to p^2 = 4, we find the roots by taking the square root of 4. This gives us two roots: p = 2 and p = -2. These roots represent the points where the quadratic equation intersects the x-axis in its graphed form. Understanding roots is crucial, as they give the possible values of the variable that solve the equation.

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