Problem 64 - Algebra A Combined Function

Question

A whole number increased by its square is two more than twice itself. Find the number.

Step-by-Step Solution

Answer

The number is 2.

Step 1: Set up the equation

Let the number be represented by the variable \( x \). The problem states that a whole number increased by its square is two more than twice itself. This can be written as an equation: \( x + x^2 = 2x + 2 \).

Step 2: Simplify the equation

Rearrange terms to make the equation easier to solve. Subtract \( 2x \) and 2 from both sides to get: \( x^2 + x - 2x - 2 = 0 \), which simplifies to \( x^2 - x - 2 = 0 \).

Step 3: Factor the quadratic equation

The equation \( x^2 - x - 2 = 0 \) can be factored into \((x - 2)(x + 1) = 0\).

Step 4: Solve for \(x\)

From the factored equation \((x - 2)(x + 1) = 0\), set each factor equal to zero: \(x - 2 = 0\) or \(x + 1 = 0\). Solving these gives the solutions \(x = 2\) and \(x = -1\).

Step 5: Verify the solution

Since we are searching for a whole number, \(x = -1\) is not suitable, as it is not a positive whole number. Verify \(x = 2\): \(2 + 2^2 = 2 + 4 = 6\) and \(2 \times 2 + 2 = 4 + 2 = 6\), which confirms \(x = 2\) satisfies the equation.

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is an essential skill in algebra, especially when trying to find the roots or solutions to a quadratic equation. Because a quadratic equation is generally in the form \[ ax^2 + bx + c = 0, \]we need to find values of \( x \) that satisfy this equation. Let's look at a straightforward way to tackle this using factoring.

• Start by making sure the quadratic equation is set to zero, i.e., all terms are on one side. This is crucial because solving equations by factoring relies on the zero product property.
• Next, express the quadratic trinomial as a product of two binomials. This is the factoring process itself. For example, in our exercise, given \( x^2 - x - 2 = 0 \), we use our knowledge of factors to write it as \((x - 2)(x + 1) = 0\).
• Check your work by expanding \((x - 2)(x + 1)\) back to confirm it equals \( x^2 - x - 2 \).

Once the quadratic is factored, each binomial represents a potential solution for\( x \). The solutions are where these values make any of the factors equal to zero.

Solving Equations

Solving equations, particularly quadratic ones, becomes truly seamless once the equation is factored. Let's delve into how we solve these after factoring:

• After factoring the quadratic equation, you'll have something like \((x - 2)(x + 1) = 0\). At this step, use the zero product property which states that if a product of factors is zero, at least one of the factors must be zero.
• This means you set each binomial equal to zero: \(x - 2 = 0\) and \(x + 1 = 0\). Solving each, you get potential solutions: \(x = 2\) and \(x = -1\).
• The solutions represent where the original quadratic equation equals zero. They’re also the points where a graph of this equation would intersect the x-axis.

This method is efficient and provides a clear pathway to find potential solutions to the problem.

Verification of Solutions

Verifying solutions is an indispensable step in solving equations. It ensures that the solutions fulfill all the conditions laid out by the original problem. Here's how verification helps secure confidence in your solutions:

• Revisit what the problem is asking for, in this case, the number that when increased by its square, equals two more than twice itself.
• Using the potential solutions found, \(x = 2\) and \(x = -1\), substitute them back into the context of the original problem.
• Since the task specifies a whole number, \(x = -1\) is invalid, leaving \(x = 2\) to check.
• Substitute: For \(x = 2\), compute \(2 + 2^2 = 6\) and confirm if it matches \(2 \times 2 + 2 = 6\). Both sides equal 6, verifying \(x = 2\) is correct.

Verification ensures that the chosen solution fits all requirements and helps avoid errors in reasoning or calculation.