Problem 8 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ 0=x^{2}+x-6 $$

Step-by-Step Solution

Answer

The solutions for the equation \(x^{2}+x-6=0\) are \(x=-3\) and \(x=2\).

Step 1: Factoring

We start with \(0 = x^{2}+x-6\). This is a quadratic equation in the form \(ax^{2} + bx + c = 0\), where \(a=1\), \(b=1\), and \(c=-6\). To factor, we are looking for two numbers that multiply to \(a*c=-6\) and add to \(b=1\). The numbers 3 and -2 satisfy both of these conditions. We rewrite the equation in factored form: \(0 = (x-2)(x+3)\).

Step 2: Setting the factors equal to zero

The solutions for the equation will be the values of \(x\) that make either \(x-2=0\) or \(x+3=0\). To find these, set each factor equal to zero and solve.

Step 3: Solve the equation \(x-2=0\)

When solving \(x-2=0\), we add 2 to each side of the equation which results in \(x=2\).

Step 4: Solve the equation \(x+3=0\)

When solving \(x+3=0\), we subtract 3 from each side of the equation which results in \(x=-3\).

Key Concepts

Factoring

Factoring is a method used to break down equations into simpler components. It is widely applied in mathematics, particularly for solving quadratic equations. In the equation \(x^2 + x - 6 = 0\), the goal is to express it as a product of two binomials. This is like finding puzzle pieces that fit perfectly together.

• First, identify the coefficients: \(a=1\), \(b=1\), and \(c=-6\).
• Next, determine two numbers that multiply to \(a \times c = -6\) and add up to \(b = 1\).
• In this case, the numbers 3 and -2 fit perfectly. Because \(3 \times -2 = -6\) and \(3 + (-2) = 1\).

These numbers allow us to factor the quadratic equation as \((x-2)(x+3)\). Factoring is like "unboxing" the mystery behind an equation, breaking it down into more manageable parts.

Solving Equations

Once an equation is factored, solving it becomes a straightforward task. The principle is simple: if a product of two terms equals zero, at least one of the terms must be zero. This is known as the zero-product property.

• Take the equation after factoring: \((x-2)(x+3) = 0\).
• Set each factor equal to zero. This gives two separate equations: \(x-2=0\) and \(x+3=0\).
• Solving these individual simple equations will provide the solutions to the original equation.

Solving equations by setting factors to zero transforms a complex problem into manageable, bite-sized pieces, making it much easier to tackle.

Quadratic Formula

The quadratic formula is an all-encompassing tool for solving quadratic equations, especially those that are difficult or impossible to factor easily. It provides a direct solution to any quadratic equation of the form \(ax^2 + bx + c = 0\). Though it's not necessary for our current problem, it's good to know! The formula is:
\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]

• \(b\) is the coefficient of \(x\).
• \(a\) and \(c\) are the coefficients of \(x^2\) and the constant term, respectively.
• The \(\pm\) symbol means there can be two potential solutions.

Using the quadratic formula ensures you won't miss any possible solutions. However, when an equation is easily factorable, this formula might become unnecessary.