Problem 33 - College Algebra

Question

Solve for the variable. $$ -(2 x)^{2}+1=-3 $$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -1\).

Step 1: Simplify the equation

Begin by simplifying the left side of the equation. The equation is:\[-(2x)^{2} + 1 = -3\]First, calculate \((2x)^{2}\) which gives us \[4x^2\].Thus, the equation becomes:\[-4x^2 + 1 = -3\].

Step 2: Isolate the variable term

Subtract 1 from both sides of the equation:\[-4x^2 = -3 - 1\].This simplifies to:\[-4x^2 = -4\].

Step 3: Divide by the coefficient of the variable term

Divide both sides by -4 to solve for \(x^2\):\[x^2 = \frac{-4}{-4}\].This simplifies to:\[x^2 = 1\].

Step 4: Solve for the variable

Take the square root of both sides to solve for \(x\):\[x = \pm \sqrt{1}\].This simplifies to:\[x = \pm 1\].

Key Concepts

Solving Algebraic Equations

To solve algebraic equations, the main objective is to find the value of the variable that makes the equation true. Algebraic equations can range from simple linear equations to more complex types like quadratic equations. The equation given involves simplification, rearranging terms, and applying basic arithmetic operations.

The general approach to solving an algebraic equation involves a few steps:

• Simplify each side if needed.
• Apply inverse operations to move terms and simplify the equation.
• Isolate the variable to find its value.

In our exercise, simplifying the left side and then using inverse operations helped us isolate the term involving the variable, allowing us to then solve for it.

Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). They are called quadratic because they involve terms where the variable is squared. These types of equations generally have up to two solutions.

In the example provided, we dealt with a part of a quadratic equation when we simplified \(-(2x)^2 + 1 = -3\) to \(-4x^2 + 1 = -3\) and finally to \(x^2 = 1\). The quadratic component \(x^2\) indicates that there might be two potential values for \(x\) requiring further solving steps, such as taking square roots.

Exponentiation

Exponentiation is a mathematical operation, involving raising a base number to an exponent power. In any expression of the form \(a^n\), \(a\) is the base, and \(n\) is the exponent.

In the original exercise, exponentiation was performed on the term \((2x)^2\), resulting in \((2)^2 \cdot (x)^2 = 4x^2\). This operation is crucial in expanding and simplifying expressions, so you can reach the form that allows solving the rest of the equation.

Isolate the Variable

Isolating the variable means rearranging the equation such that the variable is by itself on one side of the equation, making it easier to solve. This often involves

• Adding or subtracting terms to both sides of the equation.
• Dividing or multiplying both sides by a coefficient.

This method was used when the equation \(-4x^2 + 1 = -3\) was simplified to \(-4x^2 = -4\) by subtracting 1 from both sides. By using these steps, we eventually isolated \(x^2\) and solved for \(x\) by taking the square root, yielding solutions \(x = \pm 1\).