Problem 58 - Algebra 2

Question

Solve each equation \(d+\sqrt{d}-132=0\)

Step-by-Step Solution

Answer

The solution is \( d = 121 \).

Step 1: Move Constant to Right Side

First, we need to isolate terms involving the variable on one side. Subtract 132 from both sides of the equation to achieve this:\[ d + \sqrt{d} = 132 \]

Step 2: Substitute Variable

To simplify, let's substitute \( \sqrt{d} \) with a new variable, say \( x \). Thus, \( x^2 = d \). The equation now becomes:\[ x^2 + x = 132 \]

Step 3: Form a Quadratic Equation

By rearranging, we have:\[ x^2 + x - 132 = 0 \]This is now a standard quadratic equation.

Step 4: Solve Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), and \( c = -132 \). Calculate the discriminant:\[ b^2 - 4ac = 1^2 - 4 \times 1 \times (-132) = 1 + 528 = 529 \]Now solve using the quadratic formula:\[ x = \frac{-1 \pm \sqrt{529}}{2} \]Since \( \sqrt{529} = 23 \), the solutions are:\[ x = \frac{-1 + 23}{2} = 11 \] and \[ x = \frac{-1 - 23}{2} = -12 \]

Step 5: Determine Values of \( d \)

Recall that \( x = \sqrt{d} \), meaning \( x^2 = d \). Thus, given the solutions for \( x \):- \( x = 11 \): \( d = 11^2 = 121 \)- \( x = -12 \): \( d = (-12)^2 = 144 \)However, \( \sqrt{d} \) represents the principal, non-negative root, so \( x = -12 \) does not apply. Hence, \( d = 121 \) is the only valid solution.

Key Concepts

Substitution Method

The substitution method is a powerful tool that simplifies the solving process by introducing a new variable. This technique effectively transforms complex equations into a more manageable form.
Imagine we have the equation \( d + \sqrt{d} = 132 \). The term \( \sqrt{d} \) might complicate direct solving. Therefore, we substitute \( \sqrt{d} \) with a variable, say \( x \). This turns our problem into \( x^2 + x = 132 \). As a result, we now handle a simple quadratic equation instead of one involving roots.
Substitution serves as a bridge, translating challenging expressions into ones we're more accustomed to solving.

Quadratic Formula

The quadratic formula is a universal method to find solutions of any quadratic equation.
Given a standard quadratic form \( ax^2 + bx + c = 0 \), the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) will provide solutions for \( x \) when \( a eq 0 \).
In our exercise with the equation \( x^2 + x - 132 = 0 \), we have \( a = 1 \), \( b = 1 \), and \( c = -132 \).
By plugging these into our quadratic formula, we derive the solutions:

• \( x = \frac{-1 + 23}{2} = 11 \)
• \( x = \frac{-1 - 23}{2} = -12 \)

The quadratic formula saves time and eliminates guesswork, giving you exact solutions efficiently.

Discriminant

The discriminant of a quadratic equation, denoted as \( b^2 - 4ac \), signals the nature of the solutions.
It tells us how many solutions the quadratic equation has and whether they are real or complex.
For the equation \( x^2 + x - 132 = 0 \), the discriminant is calculated as follows:

• \( b^2 - 4ac = 1^2 - 4 \times 1 \times (-132) = 529 \)

Since 529 is a positive perfect square, it indicates two distinct real solutions. This was evident when we found \( x = 11 \) and \( x = -12 \).
Thus, understanding the discriminant helps predict the solution's nature before fully solving the equation.

Principal Roots

Principal roots refer to the primary, non-negative solutions of a square root expression.
In mathematical problems, especially when dealing with squares and their roots, we often consider this aspect.
In our exercise, although the solutions for \( x \) are \( 11 \) and \( -12 \), only \( x = 11 \) corresponds to a valid \( \sqrt{d} \). This is because the principal root conventionally ignores negative values.
Hence, by recalling that \( x = \sqrt{d} \), only the non-negative outcome leads to \( d = 11^2 = 121 \).
Understanding principal roots is crucial as it refines answers to align with real-world interpretations and expectations.