Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=3 x+18$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Equation

The original equation is given as \(x^{2} = 3x + 18\). First, we need to rewrite it in a standard quadratic form, which is \(ax^2 + bx + c = 0\). To do this, subtract \(3x\) and \(18\) from both sides to get \(x^{2} - 3x - 18 = 0\).

Step 2: Factor the Quadratic Equation

To factor \(x^{2} - 3x - 18 = 0\), we look for two numbers that multiply to \(-18\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). These numbers are \(-6\) and \(3\). Therefore, the factored form is \((x - 6)(x + 3) = 0\).

Step 3: Solve for x

Using the factored form \((x - 6)(x + 3) = 0\), set each factor equal to zero: \(x - 6 = 0\) and \(x + 3 = 0\). Solving these, we get \(x = 6\) and \(x = -3\). These are the solutions to the equation.

Step 4: Graph the Solutions

To graph the solutions, plot the quadratic equation \(y = x^{2} - 3x - 18\). The parabola will intersect the x-axis at the solutions \(x = 6\) and \(x = -3\), which confirms graphically that these are the correct solutions.

Key Concepts

Factoring Quadratics

Factoring quadratics can seem daunting at first, but it's a vital skill when solving quadratic equations. The essence of factoring is to rewrite the quadratic expression as a product of two binomials. For an equation in the form of \( ax^2 + bx + c = 0 \), you are looking for two numbers that multiply to \( ac \) (the product of the coefficient of \(x^2\) and the constant term \(c\)) and add up to \( b \), the coefficient of \(x\).

In the exercise provided, the equation \( x^2 - 3x - 18 = 0 \) was factored into \((x - 6)(x + 3) = 0\). This is because:

• The product needed is \(-18\), which is \(1\times -18\).
• The sum needed is \(-3\).

The numbers \(-6\) and \(3\) satisfy both conditions, making them the right choice. Factoring is essentially unwrapping the expression to understand the values of \(x\) that satisfy the equation.

Graphing Parabolas

Graphing a parabola can provide a visual confirmation of your solutions to a quadratic equation. When graphed, a quadratic equation will produce a U-shaped curve known as a parabola. The general form of a quadratic equation is \( y = ax^2 + bx + c \). Understanding the features of the parabola—such as the vertex, axis of symmetry, and roots—is crucial.

The exercise asks for graphing \( y = x^2 - 3x - 18 \). The parabola will intersect the x-axis at the roots \( x = 6 \) and \( x = -3 \), where the expression equals zero. These points on the x-axis are called the 'solutions', 'roots', or 'zeros' of the equation.

• The vertex of the parabola provides the maximum or minimum point and is a key feature when sketching.
• The axis of symmetry goes through the vertex, dividing the parabola into two mirror-image halves.

These elements guide the sketching and help in verifying the solutions found algebraically.

Solving Equations

Solving equations, especially quadratic ones, involves finding the values of the variable that make the equation true. Once you have represented a quadratic equation in its standard form \( ax^2+bx+c=0 \), you can solve it through factoring, using the quadratic formula, or completing the square.

For the provided problem, the method of factoring was used. After factoring the equation into \((x - 6)(x + 3) = 0\), the solutions are found by setting each factor to zero and solving for \(x\):

• \( x - 6 = 0 \) results in \( x = 6 \).
• \( x + 3 = 0 \) results in \( x = -3 \).

These values satisfy the original equation, confirming them as correct solutions. Solving equations is about isolating the variable and finding its potential values, which makes each step crucial in reaching the final solution.