Problem 23 - Algebra 1: Concepts and Skills

Question

Use a graph to estimate the solutions of the equation. Check your solutions algebraically. $$-4 x^{2}-8 x=-12$$

Step-by-Step Solution

Answer

The solutions to the equation are x = 1 and x = 3.

Step 1: Rewrite the Equation

The given equation is \(-4x^2 - 8x = -12\). It can be rewritten by adding 12 to both sides, which yields \(-4x^2 - 8x + 12 = 0\).

Step 2: Find Roots Algebraically

Solve the equation \(-4x^2 - 8x + 12 = 0\) for x using the quadratic formula, which gives \(x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4*-4*12}}{2*-4}\). Calculating the values gives two possible solutions for x, namely 1 and 3.

Step 3: Verify Graphically

To verify the roots graphically, plot the function \(-4x^2 - 8x + 12\) using a suitable plotting tool. The points where the graph meets the x-axis represent the solutions to the equation. If the graph touches x-axis at x = 1 and x = 3, it would confirm the results.

Step 4: Compare Solutions

Compare the results of the graphical method and the algebraic method. If they match, it means that the solutions have been estimated correctly.

Key Concepts

Graphical Solution of Equations

Finding a solution to an equation graphically involves plotting the curve of the equation on a coordinate system to visually determine where it intersects the x-axis. These points of intersection represent the solutions or "roots" of the equation.

Let's break down the process:

• Rewrite the given equation in standard quadratic form, if not already done. This involves moving all terms to one side of the equation, making it equal to zero. For example, turning \(-4x^2 - 8x = -12\) into \(-4x^2 - 8x + 12 = 0\).
• Plot the corresponding graph of the quadratic equation. Tools range from graph paper to digital graphing calculators or software programs.
• Identify where the curve touches or crosses the x-axis. These are the graphical solutions or estimated roots of the equation.

By visually examining the graph, one can confirm the algebraic solutions and gain insight into the nature of the equation, such as the number of real roots. This represents a practical and intuitive method of verifying solutions.

Quadratic Formula

The quadratic formula is a crucial tool for solving quadratic equations, offering a systematic approach to finding the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given as:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]
It offers both potential solutions to the equation in terms of the coefficients \(a\), \(b\), and \(c\). This formula is especially handy when the quadratic is not easily factorable. Here's how you can use it:

• Identify the values of \(a\), \(b\), and \(c\) from your equation. For instance, in \(-4x^2 - 8x + 12 = 0\), we have \(a = -4\), \(b = -8\), and \(c = 12\).
• Substitute these values into the quadratic formula.
• Compute the solutions by evaluating the expression. The discriminant \(b^2 - 4ac\) here is important as it indicates the nature of the roots. A positive discriminant implies two real solutions, zero implies one real solution, and a negative discriminant indicates no real solutions.

With accurate numerical calculation, the quadratic formula provides precise solutions, as seen in solving the equation \(x = 1\) and \(x = 3\) for \(-4x^2 - 8x + 12 = 0\).

Roots of Equations

In the context of quadratic equations, roots, or zeroes, are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). They are the points where the graph of the quadratic equation intersects the x-axis.

Understanding and finding the roots of an equation is fundamental in solving quadratic equations as it delivers specific solutions. Here's what to consider:

• Quadratic equations can have a maximum of two roots, since the degree of the polynomial is two. These roots can be real or complex numbers.
• The discriminant \((b^2 - 4ac)\) in the quadratic formula plays a pivotal role in determining the nature of these roots.
• If the discriminant is positive, the equation has two distinct real roots. If zero, there is exactly one real root, known as a repeated or double root. A negative discriminant implies two complex roots.

For the equation \(-4x^2 - 8x + 12 = 0\), using either the quadratic formula or graphically determining, we find the roots \(x = 1\) and \(x = 3\). These roots confirm that the solutions meet both algebraic and graphical verifications, affirming the interconnectedness of different solving methods.