Problem 13 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by the zero-factor property. $$x^{2}-5 x+6=0$$

Step-by-Step Solution

Answer

x = 2 or x = 3

Step 1 - Write the equation in standard form

The given equation is already in standard form: \[x^2 - 5x + 6 = 0.\]

Step 2 - Factor the quadratic expression

Factor the quadratic expression on the left-hand side of the equation: \[x^2 - 5x + 6 = (x - 2)(x - 3).\]

Step 3 - Apply the zero-factor property

Set each factor equal to zero using the zero-factor property: \[x - 2 = 0 \text{ or } x - 3 = 0.\]

Step 4 - Solve each equation

Solve each of the simple equations: \[x = 2 \text{ or } x = 3.\]

Key Concepts

solving quadratic equations

Quadratic equations are equations of the form
\(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. When solving these equations, the goal is to find the values of \(x\) that make the equation true. These values are known as the roots of the quadratic equation.
There are several methods to solve quadratic equations:

• Factoring
• Using the quadratic formula
• Completing the square

For the given equation,
\(x^2 - 5x + 6 = 0\),
, factoring is a suitable and straightforward method.

factoring

Factoring is a technique used to write a polynomial as a product of its simpler polynomials. To factor a quadratic equation, like
\(x^2 - 5x + 6 = 0\),
you need to find two numbers that multiply to the constant term (\(c\)) and add up to the coefficient of the linear term (\(b\)).

For the equation \(x^2 - 5x + 6 = 0\), we need to find two numbers that multiply to \(6\) and add up to \(-5\). These numbers are \(-2\) and \(-3\).

Therefore, the quadratic equation can be factored as
\((x - 2)(x - 3) = 0\).

standard form of a quadratic equation

To solve a quadratic equation, it must first be written in its standard form:
\( ax^2 + bx + c = 0\).
In standard form,
a quadratic equation consists of:

• The leading term, \(ax^2\), where \(a\) is the coefficient of the quadratic term.
• The linear term, \(bx\), where \(b\) is the coefficient of the linear term.
• The constant term, \(c\).

In our example,
\(x^2 - 5x + 6 = 0\),
the equation is already in standard form, with:

• \(a = 1\)
• \(b = -5\)
• \(c = 6\)

roots of quadratic equations

The roots of a quadratic equation are the values of \(x\) that satisfy the equation. These roots are also known as the solutions of the quadratic equation. After factoring a quadratic equation, we can use the zero-factor property to find its roots.

The zero-factor property states that if the product of two factors is zero, then at least one of the factors must be zero. For the equation
\((x - 2)(x - 3) = 0\),
we set each factor to zero:
- \(x - 2 = 0\) - This gives \(x = 2\)
- \(x - 3 = 0\) - This gives \(x = 3\)

Therefore, the roots of the quadratic equation \(x^2 - 5x + 6 = 0\) are \(x = 2\) and \(x = 3\).