Problem 54 - Elementary Algebra

Question

A positive real number is 1 more than twice another. If 4 times the smaller number is subtracted from the square of the larger, then the result is 21 . Find the numbers. Geometry Problems Round off your answers to the nearest hundredth.

Step-by-Step Solution

Answer

The smaller number is 2.24 and the larger number is 5.48.

Step 1: Define Variables

Let the smaller number be represented by \( x \). We know from the problem that the larger number is 1 more than twice the smaller number, so we write the larger number as \( 2x + 1 \).

Step 2: Interpret the Condition

We are given that 4 times the smaller number subtracted from the square of the larger number equals 21. This can be written as the equation: \((2x + 1)^2 - 4x = 21\).

Step 3: Expand the Equation

Expand the left side of the equation: \((2x + 1)^2\) becomes \(4x^2 + 4x + 1\). The equation now looks like: \(4x^2 + 4x + 1 - 4x = 21\).

Step 4: Simplify the Equation

Simplify the equation by combining like terms: \(4x^2 + 1 = 21\).

Step 5: Solve the Quadratic Equation

Subtract 1 from both sides to make it \(4x^2 = 20\). Then divide each side by 4 to isolate \(x^2\): \(x^2 = 5\).

Step 6: Find the Value of x

Solve for \(x\) by taking the square root of both sides: \(x = \sqrt{5}\).\( \) approximates to 2.24.

Step 7: Calculate the Larger Number

Substitute \(x = \sqrt{5}\) into \(2x + 1\) to find the larger number: \(2(2.24) + 1 = 4.48 + 1 = 5.48\).

Step 8: Verify Your Solution

Verify by checking that \( (5.48)^2 - 4(2.24) = 21 \). Simplifying gives \( 30.07 - 8.96 \approx 21 \), confirming the solution.

Key Concepts

Quadratic Equations

Quadratic equations are a crucial part of algebra that can describe many real-world situations. They typically come in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable we need to solve for. These equations can have up to two solutions because they are derived from a parabola intersecting the x-axis.

In this exercise, the equation \((2x + 1)^2 - 4x = 21\) was rearranged and simplified to the quadratic form \( 4x^2 = 20 \). Finding a solution involves isolating the \( x^2 \) term and then solving for \( x \). This approach allows us to determine the value of \( x \) accurately, which represents a key feature of quadratic equations - their ability to model situations where variables squared lead to real answers.

Real Numbers

In algebra, real numbers are all the numbers that we typically work with on a daily basis, including whole numbers, fractions, decimals, and irrational numbers like \( \sqrt{2} \). They are the numbers that you can place on a number line, extending infinitely in both positive and negative directions.

The problem mentions finding positive real numbers which play a critical role in defining the parameters of this scenario. Here, when we solve the quadratic equation and arrive at \( x = \sqrt{5} \), we are engaged in finding a real number solution. This result is a positive real number, and the larger number calculated as \( 5.48 \) also lies comfortably within the realm of real numbers.

Variable Substitution

Variable substitution is a method used for simplifying problems, especially algebraic equations, by replacing complex expressions with single variables.

In this problem, the smaller number is chosen to be the variable \( x \). This helps in rewriting expressions and performing calculations more easily.

• We initially let the smaller number be \( x \).
• The larger number is expressed as \( 2x + 1 \).

This substitution helps us directly write and simplify the equation \((2x + 1)^2 - 4x = 21\), without getting confused with the longer expression. Simplification through substitution makes complex algebraic manipulations more manageable and straightforward.

Problem-Solving Steps

Solving algebraic problems step-by-step helps to avoid errors and ensure clarity. Systematic approaches turn complex tasks into manageable ones and guide us from problem statement to solution.

• Start by defining your variables: decide which values you'll use to represent unknowns.
• Interpret conditions and constraints as algebraic equations.
• Expand and simplify the equation as necessary.
• Break down the simplified problem by solving for unknown values.
• Once a solution is found, verify it through substitution back into the original equation.

In this case, we clearly defined the relationship between two numbers, created a quadratic equation, simplified it to solve for \( x \), and verified our results, confirming their correctness.