Problem 45 - Elementary Algebra

Question

Solve. $$3 x=1 x+1+13 x(x+1)$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{-11 + \sqrt{69}}{26}\) and \(x = \frac{-11 - \sqrt{69}}{26}\).

Step 1: Simplify the Equation

Let's start by simplifying the given equation \(3x = 1x + 1 + 13x(x+1)\). First, separate the terms on each side: \(3x\) on the left, and \(1x + 1 + 13x^2 + 13x\) on the right. Combine like terms on the right: \(1x + 13x = 14x\), giving us \(3x = 13x^2 + 14x + 1\).

Step 2: Set the Equation to Zero

Next, we want to bring all terms to one side to set the equation to zero. Subtract \(3x\) from both sides to get: \[0 = 13x^2 + 14x + 1 - 3x\].Simplify the expression: \[0 = 13x^2 + 11x + 1\].

Step 3: Solve for x Using the Quadratic Formula

Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 13\), \(b = 11\), and \(c = 1\). Calculate the discriminant: \(b^2 - 4ac = 11^2 - 4 \times 13 \times 1 = 121 - 52 = 69\).Compute the solutions: \[x = \frac{-11 \pm \sqrt{69}}{26}\].

Step 4: Finalize the Answer

We found that the solutions are \[x = \frac{-11 + \sqrt{69}}{26}\] and \[x = \frac{-11 - \sqrt{69}}{26}\]. These are the solutions in their exact form.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. These are equations that take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The formula helps us find the values of \(x\) that satisfy the equation. It is expressed as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula may appear daunting at first, but it becomes straightforward with practice.When solving a quadratic equation using the quadratic formula:

• Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
• Compute the discriminant \(b^2 - 4ac\), which will help you determine the nature of the roots.
• Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula to find the solutions for \(x\).

The "±" symbol indicates that there are typically two solutions: one for the addition and one for the subtraction.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and carrying out arithmetic operations to rewrite the expression in a simpler form. It's like cleaning up your workspace so everything is tidy and organized.In the provided solution, the original equation is:

• \(3x = 1x + 1 + 13x(x+1)\).

Here's what simplifying entails: - Combine the terms that have the same variables, like \(1x\) and \(13x\). This combines to form \(14x\).- The equation can then be rearranged to get \(3x = 13x^2 + 14x + 1\) once the expression \(13x(x+1)\) is expanded and added to the equation.Simplifying helps in rearranging the equation into the standard quadratic form \(ax^2+bx+c=0\), making it perfect for using the quadratic formula. A tidy equation is easier to work with for further calculations.

Discriminant

The discriminant plays a crucial role in determining the nature of the solutions of a quadratic equation. It is the part of the quadratic formula under the square root: \(b^2 - 4ac\). Here's why the discriminant is crucial:

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root (also called a repeated or double root).
• If the discriminant is negative, the equation has no real roots. Instead, it has two complex conjugate roots.

In our exercise, after substituting \(b = 11\), \(a = 13\), and \(c = 1\):- We calculated the discriminant as \(69\), a positive number.This means our quadratic equation has two distinct real roots, which we can find using the formula. Knowing this helps confirm that our approach to solving the equation was correct and guides us on what type of answers to expect.