Problem 61 - Elementary Algebra

Question

Solve. $$ (x+1)(6 x+1)=2 x $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{-1}{3}\) and \(x = \frac{-1}{2}\).

Step 1: Expand the Expression

First, expand the left side of the equation by using the distributive property: \((x+1)(6x+1) = x \cdot (6x+1) + 1 \cdot (6x+1) = 6x^2 + x + 6x + 1 = 6x^2 + 7x + 1\). The equation becomes:\[6x^2 + 7x + 1 = 2x\].

Step 2: Bring All Terms to One Side

Subtract \(2x\) from both sides to set the equation to zero:\[6x^2 + 7x + 1 - 2x = 0\] which simplifies to \[6x^2 + 5x + 1 = 0\].

Step 3: Identify the Quadratic Equation Parameters

In the quadratic equation \(ax^2 + bx + c = 0\), identify \(a = 6\), \(b = 5\), and \(c = 1\).

Step 4: Use the Quadratic Formula

To solve the quadratic equation, use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Substitute \(a\), \(b\), and \(c\) into the formula:\[x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 6 \cdot 1}}{2 \cdot 6}\].

Step 5: Calculate the Discriminant

Calculate the discriminant, \(b^2 - 4ac\):\(b^2 - 4ac = 5^2 - 4 \cdot 6 \cdot 1 = 25 - 24 = 1\).

Step 6: Find the Roots

Substitute the discriminant back into the quadratic formula:\[x = \frac{-5 \pm \sqrt{1}}{12}\]. This gives two solutions: \[x = \frac{-5 + 1}{12} = \frac{-4}{12} = \frac{-1}{3}\] and \[x = \frac{-5 - 1}{12} = \frac{-6}{12} = \frac{-1}{2}\].

Step 7: Finalize the Solutions

The solutions to the equation are \(x = \frac{-1}{3}\) and \(x = \frac{-1}{2}\). Confirm by substituting back into the original equation to verify.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra, used to simplify expressions and solve equations. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. This can be shown as:

• For any numbers or expressions, if you have an expression like \((a+b)c\), you can distribute the multiplication over addition, getting \(ac + bc\).

In the context of solving quadratic equations like \((x+1)(6x+1) = 2x\), the distributive property helps to transform the expression by combining all terms together. This was done in Step 1 of the solution, where the terms \(x \cdot (6x+1) + 1 \cdot (6x+1)\) were expanded to become \(6x^2 + 7x + 1\). This step is crucial because it simplifies complex expressions into a standard quadratic format.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides the roots directly, saving time compared to factoring, especially when the equation does not easily factorize. Whenever dealing with any quadratic equation, first identify the coefficients \(a\), \(b\), and \(c\). Insert these values into the formula to calculate the solutions.In our exercise, we have \(a=6\), \(b=5\), and \(c=1\). Substituting these into the formula, we find the roots of the quadratic equation. It helps solve for \(x\) even in cases where the roots are not rational numbers, covering virtually every kind of quadratic equation.

Solving Equations

Solving equations is at the heart of algebra. For quadratic equations, this involves a few crucial steps to find their roots. Breaking them down:

• Bring all terms to one side: This involves rearranging the terms so that one side of the equation equals zero, making it possible to apply the quadratic formula.
• Simplify: Combine like terms. This was seen when \(2x\) was moved to the left side in our exercise, leading to \(6x^2 + 5x + 1 = 0\).

Once in a simplified form, techniques like the quadratic formula can be employed to find solutions. Quadratic equations can have zero, one, or two solutions, depending on the value of the discriminant. This systematic approach helps simplify and solve even the most complex equations.

Discriminant

The discriminant is a component of the quadratic formula that tells us about the nature of the roots of a quadratic equation. It's represented as \(b^2 - 4ac\). The discriminant has crucial significance:

• Positive Discriminant: Indicates two real and distinct roots.
• Zero Discriminant: Yields exactly one real root, i.e., the roots are equal.
• Negative Discriminant: Suggests two complex (imaginary) roots.

In the given problem, the discriminant is calculated as \(5^2 - 4 \cdot 6 \cdot 1 = 1\), a positive number. This reveals that the equation has two distinct real roots. Understanding the discriminant allows us to predict the outcome of the quadratic equation before even solving it fully, helping us frame our expectations about the solutions.