Problem 26 - Algebra 2

Question

Solve each equation by factoring. Then graph. \(x^{2}-4 x=21\)

Step-by-Step Solution

Answer

Solutions are \(x = 7\) and \(x = -3\). Graph the parabola with these x-intercepts.

Step 1: Rewrite the Equation

First, rewrite the given equation in the standard quadratic form. The equation is given as \(x^{2} - 4x = 21\). To write it in standard form, bring all terms to one side of the equation. This results in \(x^{2} - 4x - 21 = 0\).

Step 2: Factor the Quadratic Expression

Next, factor the quadratic expression \(x^{2} - 4x - 21\). We need two numbers that multiply to \(-21\) and add to \(-4\). These numbers are \(-7\) and \(3\). Thus, the factorized form is \((x - 7)(x + 3) = 0\).

Step 3: Solve for x

Set each factor equal to zero to solve for \(x\). For \(x - 7 = 0\), solving gives \(x = 7\). For \(x + 3 = 0\), solving gives \(x = -3\). Thus, the solutions are \(x = 7\) and \(x = -3\).

Step 4: Graph the Solutions

Plot the solutions \(x = 7\) and \(x = -3\) on the x-axis. Since the original equation was quadratic, the graph is a parabola that opens upwards. The vertex of the parabola is at the midpoint of the solutions (\(x = 2\)), and it crosses the y-axis at the term \(c = -21\) of the quadratic equation \(x^{2} - 4x - 21 = 0\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a key skill for solving quadratic equations. The idea is to express a quadratic equation like \(ax^2 + bx + c = 0\) as a product of two binomials. For instance, with the equation \(x^2 - 4x - 21 = 0\), we want to find two numbers that multiply to the constant term (-21) and add to the linear coefficient (-4). This approach is helpful because it allows us to break down the equation into simpler terms, making it easier to find the values of \(x\) that solve the equation.

In our example, these numbers are -7 and 3. Thus, we can rewrite the quadratic as \((x - 7)(x + 3) = 0\). This is called its factored form. To find the solutions for \(x\), we set each factor equal to zero: \(x - 7 = 0\) and \(x + 3 = 0\). Solving these equations gives \(x = 7\) and \(x = -3\) as the solutions.

• Look for factors that multiply to \(c\) (the constant term).
• Check if they add up to \(b\) (the coefficient of \(x\)).
• Use these numbers to write as factors: \((x - p)(x + q)\), where \(p\) and \(q\) fulfill both conditions.

Quadratic Functions

Quadratic functions have the general form \(f(x) = ax^2 + bx + c\) and are characteristic for their symmetric curved shape called a parabola. A key feature of these functions is their vertex, which is the peak or the lowest point of the parabola, depending on whether it opens downward or upward.

In solving quadratic equations by factoring, we often obtain a function in a factorized form \((x - p)(x + q) = 0\). Each factor provides an important feature: the points \(p\) and \(q\) are the x-intercepts of the graph where the function equals zero.

This function is essential not only for solving quadratic equations but also for graphically representing real-world phenomena like projectile motion, which follows a parabolic trajectory.

• Vertex is crucial for understanding the parabola's direction and position.
• X-intercepts (the solutions) are where the function touches the x-axis.

Graphing Parabolas

Graphing parabolas involves plotting the function as a curved line on a coordinate grid. For the quadratic equation \(x^2 - 4x - 21 = 0\), once factored (\((x - 7)(x + 3) = 0\)), the solutions \(x = 7\) and \(x = -3\) tell us where the parabola crosses the x-axis. This gives us accurate starting points for sketching the parabola.

To graph the parabola:

• First, plot the x-intercepts (solutions) on the x-axis.
• Identify the vertex, which lies halfway between the x-intercepts, at \(x = \frac{7 + (-3)}{2} = 2\).
• The y-intercept is found by setting \(x = 0\) in \(f(x) = x^2 - 4x - 21\), resulting in \(f(0) = -21\).

These points allow you to sketch the parabola accurately. Understanding how these features (vertex, x-intercepts, y-intercept) define the parabola's shape enables you to predict behavior in varying contexts.