Problem 1 - The Calculus with Analytic Geometry

Question

x^{2}=-y ; y=-4

Step-by-Step Solution

Answer

x = 2 or x = -2

Step 1: Substitute the Given Value

Substitute the given value of y into the equation. Since y = -4, replace y in the equation with -4. This gives: \[ x^2 = -(-4) \]

Step 2: Simplify the Equation

Simplify the equation by performing the arithmetic operation. \[ x^2 = 4 \]

Step 3: Solve for x

To solve for x, take the square root of both sides. Remember that the square root of a number can be positive or negative. \[ x = \pm \sqrt{4} \]Therefore: \[ x = 2 \]or\[ x = -2 \]

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. A quadratic equation is typically written in the form ) or ) The goal is to find the values of x that make the equation true. In this example: -Start by substituting the known values. With y = -4, we replace y in the equation with -4 to get: -Simplify the equation by performing any required arithmetic. -Notice that to isolate x, we undo the squared term by taking the square root of both sides. It's crucial to know that taking a square root introduces both positive and negative solutions. Hence, the solutions to are x = 2 and x = -2.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For any non-negative number a, the principal square root of a is denoted by Notice that a square root has two possible values, a positive root and a negative root, since both squares of positive and negative numbers are the same. For example, consider: The number 4 has two square roots: 2 and -2. When solving quadratic equations, remember that both the positive and negative roots must be considered to find all possible solutions. When taking the square root on both sides of the equation always write the solutions as: Thus, and are explicit solutions.

Arithmetic Operations

Arithmetic operations are fundamental mathematical operations used to perform calculations. They include:

• Addition (+)
• Subtraction (-)
• Multiplication (×)
• Division (÷)

When simplifying an equation, it is often necessary to use multiple arithmetic operations. Let's look at our exercise example in steps:

• Step 1: Substituting y = -4 into the equation yields:
• Step 2: Perform arithmetic operations of multiplication and subtraction to simplify: yields:
  • Multiplying -1 by -4 results in 4.
• Step 3: To isolate x, take the square root of both sides, which involves understanding and applying the arithmetic of square roots.

By mastering these basic operations, solving more complex mathematical problems becomes much easier and more intuitive.