Problem 47 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$(x-1)(x+4)=14$$

Step-by-Step Solution

Answer

The solutions for the equation are \(x=3\) and \(x=-6\).

Step 1: Expand the Left-Hand Side

Start by multiplying out the factors on the left-hand side of the equation. It results in \(x^2+3x-4=14\).

Step 2: Rearrange the Equation

Subtract 14 from both sides to set the equation to zero. The equation now reads \(x^2+3x-4-14=0\), which simplifies to \(x^2+3x-18=0\).

Step 3: Factor the Quadratic Equation

Now, factor the quadratic equation, \(x^2+3x-18\), which becomes \((x-3)(x+6)=0\).

Step 4: Solve for \(x\)

Setting each factor equal to zero and solving for \(x\) gives the solutions \(x=3\) and \(x=-6\).

Step 5: Verify the Solution

Check the values by substituting them back into the original equation. Indeed, \((3-1)(3+4)=14\) and \((-6-1)(-6+4)=14\), thus both solutions are correct.

Key Concepts

Quadratic Equation

A quadratic equation typically takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). It is called quadratic because "quad" means square, which relates to the highest degree or power of the variable \( x^2 \). This type of equation can have two real roots, one real root, or two complex roots. Quadratic equations can be solved using various methods, such as factoring, the quadratic formula, or completing the square. Each method is suited for different types of quadratic equations, and it's useful to understand all methods to choose the most efficient one for a given problem. In this exercise, we focus on solving by factoring.

Solving Equations

Solving equations means finding values of variables that satisfy the equation. In quadratic equations, this involves finding the values of \( x \) that make the equation equal to zero. When dealing with the exercise at hand:

• Step 1: Expand the factors on the left-hand side, \((x-1)(x+4)\). This gives us \(x^2 + 3x - 4\).
• Step 2: Rearrange so that the equation equals zero, resulting in \(x^2 + 3x - 18 = 0\).

The objective in solving quadratic equations is to express it in a factorable form. Factoring involves rewriting the quadratic as a product of two binomials. After factoring, setting each binomial equal to zero will reveal the solutions for \( x \). Each solution makes the overall equation zero, effectively solving it.

X-Intercepts

X-intercepts are points where the graph of an equation crosses the x-axis. For a function \( f(x) \), the x-intercept occurs at \( f(x) = 0 \). Thus, solving the quadratic equation gives the x-intercepts of the graph. In the given exercise:

• The quadratic equation \(x^2 + 3x - 18 = 0\) factors into \((x - 3)(x + 6) = 0\).
• Setting each factor to zero gives solutions \(x = 3\) and \(x = -6\).

These solutions \( x = 3 \) and \( x = -6 \) are the x-intercepts, meaning the graph of the equation will cross the x-axis at these points. Identifying x-intercepts is crucial for understanding the behavior of quadratic functions.

Graphing Utility

A graphing utility is a tool, either a physical calculator or software, used to visually represent equations. It can display the graph of a quadratic equation to show solutions and intercepts.To verify the solution using a graphing utility, follow these steps:

• Input the function \( f(x) = x^2 + 3x - 18 \) into the graphing tool.
• Observe where the graph intersects the x-axis, which should be at \((3,0)\) and \((-6,0)\).

These points confirm the solutions \( x = 3 \) and \( x = -6 \). Using a graphing utility makes it easy to check the accuracy of solved solutions visually and helps in understanding the function's shape and behavior.