Problem 7 - College Algebra

Question

Solve the quadratic equation by factoring. $$ x^{2}-9 x+18=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3\) and \(x = 6\).

Step 1: Identify the equation format

The given quadratic equation is in the standard form: \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = -9\), and \(c = 18\).

Step 2: Find two numbers that multiply to ac and add to b

We need to determine two numbers that multiply to \(a \cdot c = 1 \cdot 18 = 18\) and add to \(b = -9\). These numbers are \(-3\) and \(-6\) because \((-3) \cdot (-6) = 18\) and \((-3) + (-6) = -9\).

Step 3: Rewrite the middle term using the two numbers

Rewrite the quadratic equation, breaking the middle term using the numbers \(-3\) and \(-6\): \(x^2 - 3x - 6x + 18 = 0\).

Step 4: Factor by grouping

Group the terms to factor by grouping. First, pair the first two terms and the last two terms: \((x^2 - 3x) + (-6x + 18)\).

Step 5: Factor each pair

Factor out the greatest common factor from each pair. For \(x^2 - 3x\), factor out \(x\), giving \(x(x - 3)\). For \(-6x + 18\), factor out \(-6\), giving \(-6(x - 3)\). The equation becomes: \(x(x - 3) - 6(x - 3) = 0\).

Step 6: Factor out the common binomial

Notice that \((x - 3)\) is a common factor. Factor it out: \((x - 3)(x - 6) = 0\).

Step 7: Solve the equation

Set each factor equal to zero: \(x - 3 = 0\) or \(x - 6 = 0\). Solving gives \(x = 3\) or \(x = 6\).

Key Concepts

Factoring

Factoring is a technique widely used to simplify algebraic expressions, especially when solving quadratic equations. The core idea of factoring is to express the quadratic equation as a product of two binomials. In the equation \[ x^2 - 9x + 18 = 0 \] the objective is to break it down into \[ (x - a)(x - b) = 0 \] format. The benefit of this form is that any solution to the equation involves finding the values of \( a \) and \( b \) such that their product equals the constant term \( c \) and their sum equals the middle coefficient \( b \). This simplifies the process, as whatever makes each binomial zero is a solution to the equation. By factoring, you are essentially expressing the problem in a simpler form that makes finding solutions more manageable.

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. There are several methods to solve these equations, but factoring is one of the most straightforward ways when applicable.

• Step 1: Write the equation in standard form. Make sure all terms are on one side of the equal sign.
• Step 2: Find a pair of numbers that multiplies to \(a \cdot c\) and adds to \(b\). These will help in rewriting the equation accurately.
• Step 3: Use these numbers to break the middle term and factor by regrouping. This allows the quadratic equation to be expressed as a product of binomials.

Once it is factored, the solutions to the equation are found at the roots of the binomials, leading directly to the values of \(x\) that solve the equation.

Solutions to Quadratic Equations

The solutions to a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). In the context of factoring, these solutions occur where each binomial equals zero. For \[x^2 - 9x + 18 = 0\] after it is factored into \[(x - 3)(x - 6) = 0\], the solutions are found by setting each binomial equal to zero.

• The equation \(x - 3 = 0\) yields the solution \(x = 3\).
• The equation \(x - 6 = 0\) yields the solution \(x = 6\).

These values are the 'roots' or 'zeroes' of the quadratic function, reflecting the points where the quadratic graph intersects the x-axis. Understanding this helps visualize not just solving the equation but also interpreting its graphical representation.