Problem 79 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ z(z-7)=-12 $$

Step-by-Step Solution

Answer

The solutions are \( z = 3 \) and \( z = 4 \).

Step 1: Expand the Equation

Start by expanding the left side of the equation. We have \( z(z-7) \), which multiplies out to \( z^2 - 7z \). Thus, the equation becomes: \[ z^2 - 7z = -12 \].

Step 2: Rearrange into a Standard Quadratic Form

To solve the quadratic equation, rearrange all terms to one side, changing the equation into the standard quadratic form \( ax^2 + bx + c = 0 \): \[ z^2 - 7z + 12 = 0 \].

Step 3: Factor the Quadratic Equation

We need to factor the quadratic expression \( z^2 - 7z + 12 \). Look for two numbers that multiply to \( 12 \) and add to \(-7\). These numbers are \(-3\) and \(-4\). Therefore, we factor the expression as: \((z-3)(z-4) = 0\).

Step 4: Solve for the Roots

Now, set each factor equal to zero such that: \[ z-3 = 0 \] or \[ z-4 = 0 \]. Solve these equations to find the roots. Thus, \( z = 3 \) or \( z = 4 \).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is a useful skill when solving quadratic equations like the one we have: \( z^2 - 7z + 12 = 0 \). To factor a quadratic expression, you need to find two numbers that multiply to the constant term and add to the coefficient of the linear term. In this case, you're looking for two numbers that multiply to \(12\) and add up to \(-7\).These numbers are \(-3\) and \(-4\). Once you have these numbers, you can rewrite the quadratic expression as a product of two binomials:

• \((z - 3)\)
• \((z - 4)\)

This is known as factoring the quadratic expression. It breaks down the equation into simpler parts that help find the roots of the equation. When you set the factored form equal to zero, it becomes much easier to solve.

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation, especially when factoring seems difficult or impossible. A quadratic equation is generally written in the form \( ax^2 + bx + c = 0 \). The quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula may seem complex at a glance, but it essentially provides the roots of any quadratic equation by substituting values for \(a\), \(b\), and \(c\). To use the formula correctly:

• Determine the coefficients \(a\), \(b\), and \(c\) from the equation.
• Insert them into the formula and solve for \(x\).

While our original problem doesn't require using the quadratic formula, it's always good to know this method in case the equation cannot be easily factored.

Roots of an Equation

The roots of an equation are the values that satisfy the equation, making it true. For a quadratic equation, the roots are found when the equation equals zero. In our exercise: \[(z - 3)(z - 4) = 0\]To find the roots, set each factor in the equation to zero:

• \(z - 3 = 0\): solving gives \(z = 3\)
• \(z - 4 = 0\): solving gives \(z = 4\)

These roots, \(z = 3\) and \(z = 4\), are the solutions to the original equation. They represent the values of \(z\) where the expression becomes true, and in a graphical sense, they are the points where the corresponding parabolic graph intersects the x-axis (or z-axis in this context). Knowing how to find these roots is essential in solving quadratic equations effectively.