Problem 22 - Algebra 2

Question

Solve each equation by factoring. Then graph. \(x^{2}-3 x-28=0\)

Step-by-Step Solution

Answer

The solutions are \(x = -4\) and \(x = 7\). Plot these as x-intercepts.

Step 1: Identify the Quadratic Equation

The equation given is a quadratic equation: \(x^2 - 3x - 28 = 0\). This equation needs to be solved by factoring.

Step 2: Find Two Numbers That Multiply And Add

We need two numbers that multiply to \(-28\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers \(4\) and \(-7\) do this because \(4 \times (-7) = -28\) and \(4 + (-7) = -3\).

Step 3: Write the Equation in Factored Form

Using the numbers found in Step 2, we can write the equation in its factored form: \((x + 4)(x - 7) = 0\).

Step 4: Solve for x

Set each factor equal to zero and solve for \(x\): \(x + 4 = 0\) gives \(x = -4\) and \(x - 7 = 0\) gives \(x = 7\). So, the solutions are \(x = -4\) and \(x = 7\).

Step 5: Graph the Solutions on a Number Line

To graph these solutions, plot the points \(x = -4\) and \(x = 7\) on a number line. These are the x-intercepts of the parabola represented by the equation.

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations which are equations of the form \(ax^2 + bx + c = 0\). In this method, we look for two numbers that multiply together to give us the product of the leading coefficient \(a\) and the constant term \(c\). In our specific equation \(x^2 - 3x - 28 = 0\), there is no leading coefficient, so we focus on finding two numbers that multiply to \(-28\) and add up to \(-3\). This gives us the solution of \((x + 4)(x - 7) = 0\), which helps in breaking the quadratic into simpler linear factors, making it easier to find solutions for \(x\).

Factoring not only provides the solutions but also reveals essential characteristics of the graph of the quadratic, like the x-intercepts.

Graphing Solutions

Graphing solutions of a quadratic equation involves plotting its roots, otherwise known as solutions, on a graph. Once the equation is factored and the solutions are found, those points on an x-axis represent where the graph touches or crosses the x-axis. The original equation \(x^2 - 3x - 28 = 0\) factors to \((x + 4)(x - 7) = 0\). Solving this gives us two solutions: \(x = -4\) and \(x = 7\). These solutions are the points at which the graph of the quadratic equation intersects the x-axis, and they are often called 'roots' or 'zeros' of the function.

When graphed, quadratics form a parabola, which either opens upwards or downwards, depending on the sign of the leading coefficient. In our example, the parabola opens upwards.

X-Intercepts

The x-intercepts of a quadratic equation are the values of \(x\) at which the graph of the equation crosses the x-axis. These intercepts can be determined by factoring the quadratic equation and finding its zeros. In our example, the equation \(x^2 - 3x - 28 = 0\) factors to \((x + 4)(x - 7) = 0\), yielding x-intercepts at \((-4, 0)\) and \((7, 0)\).

X-intercepts are significant because they indicate the exact points where the quadratic curve meets the x-axis. They also help to define the shape and position of the parabola relative to the horizontal axis. Understanding x-intercepts is crucial in graphing quadratics and solving real-world problems that can be modeled by such equations.

Factored Form

The factored form of a quadratic equation is where the equation is expressed as a product of its linear factors. For the given equation, \(x^2 - 3x - 28 = 0\), the factored form is \((x + 4)(x - 7) = 0\). This rewritten form makes it straightforward to determine the solutions of the equation by setting each factor equal to zero.

Factored form also unlocks insights into the structure of quadratic graphs:

• The roots: The solutions \((x = -4)\) and \((x = 7)\) show where the graph intersects the x-axis.
• The symmetry of the graph: The axis of symmetry can be found halfway between the roots.
• It simplifies further calculations in algebra, such as solving inequalities or verifying other properties of quadratic functions.

Quadratic Formula

The quadratic formula is another powerful tool for finding the solutions of any quadratic equation, particularly when factorization is not straightforward. The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula will work in all scenarios where you have a quadratic equation of the form \(ax^2 + bx + c = 0\). Unlike factoring, the quadratic formula provides an exact solution, regardless of whether the roots are real, imaginary, or irrational.

Using the quadratic formula involves calculating the discriminant \(b^2 - 4ac\), which reveals information about the nature of the roots:

• If it's positive, there are two distinct real roots.
• If it's zero, there is exactly one real root.
• If it's negative, the solutions are complex numbers.

While for the equation \(x^2 - 3x - 28 = 0\) factoring is simple, for more complex equations, the quadratic formula is invaluable for finding solutions.