Problem 43 - Algebra 1

Question

Represent the solution graphically. Check the solution algebraically. $$ -7 x^{2}-21 x=14 $$

Step-by-Step Solution

Answer

The solution to the quadratic equation \( -7 x^{2}-21 x - 14 = 0 \) can be found by Quadratic formula. The roots can be represented graphically as the x-intercepts of the graph of the equation. The solution can be checked algebraically by substituting the root values back into the original equation.

Step 1: Convert equation to standard form

A quadratic equation is in the standard form \( ax^{2} + bx + c = 0 \). Here, convert \( -7 x^{2}-21 x = 14 \) to standard form by bringing 14 to the left side: \( -7 x^{2}-21 x - 14 = 0 \)

Step 2: Find the roots using Quadratic Formula

The Quadratic formula is \( x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \). Here, a=-7, b=-21, c=-14. Apply these values in the Quadratic formula to get the roots.

Step 3: Graphical Representation of the solution

Plot the given equation \( -7 x^{2}-21 x - 14 = 0 \) on the graph. The points where the graph cut the X-axis are the real roots of the equation.

Step 4: Check the Solution Algebraically

Substitute the obtained roots back into the original equation and verify if both sides balance. If they do, then the roots obtained are correct, otherwise, there might be some calculations mistake.

Key Concepts

Standard Form of a Quadratic Equation

Understanding the standard form of a quadratic equation is essential for solving these types of problems. All quadratic equations can be written in the standard form, which is represented as \( ax^{2} + bx + c = 0 \) where \( a \) is the coefficient of \( x^{2} \) and must be nonzero, \( b \) is the coefficient of \( x \) and \( c \) is the constant term.

In our exercise, the given equation \( -7x^{2} - 21x = 14 \) must first be rewritten into the standard form. This involves moving all the terms to one side of the equation, yielding \( -7 x^{2}-21 x - 14 = 0 \) where \( a = -7 \) , \( b = -21 \) , and \( c = -14 \) .

Here's how this format aids the solving process:

• It sets the stage for applying various solution techniques like factoring, completing the square, or using the quadratic formula.
• It helps to classify the equation readily by identifying the coefficients and the constant term.

By keeping the equation in this standardized form, we can streamline the solving process and ensure consistency when applying different methods to find the roots of the quadratic equation.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations when other methods, like factoring, are difficult to apply.

The generic formula is \( x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \) which provides the solutions to any quadratic equation \( ax^{2} + bx + c = 0 \). The symbol \( \pm \) indicates that there will usually be two solutions, corresponding to the two possible values of \( x \) obtained by using both the positive and negative square roots.

For the exercise equation \( -7x^{2} - 21x - 14 = 0 \) , we identified \( a = -7 \) , \( b = -21 \) , and \( c = -14 \). Plugging these into the quadratic formula gives us the roots or solutions of the equation.

This method reduces potential errors and provides a systematic approach to handle even complex quadratic equations easily.

Graphical Representation of Equations

A graphical representation of a quadratic equation can provide a visual understanding of the solutions or roots of the equation.

By plotting the equation on a graph, the curve we obtain is a parabola. The points where this parabola intersects the x-axis represent the real roots of the equation. These are the same roots we found algebraically using the quadratic formula. In the context of our exercise, the graph of \( -7x^{2} - 21x - 14 = 0 \) will show the intersection points on the x-axis, revealing the solutions graphically.

Here are a few benefits of graphical representation:

• It confirms the number of real solutions: if the parabola crosses the x-axis twice, there are two real solutions; if it just touches the x-axis, there's one real solution; and if it doesn't touch the x-axis at all, there are no real solutions (they are complex numbers).
• It provides a visual check for the algebraic solutions: plotting the solutions on the graph can help verify their accuracy.

Thus, combining the algebraic approach with graphical representation ensures a deeper understanding of the concept of quadratic equations.