Problem 41 - Algebra 1

Question

Represent the solution graphically. Check the solution algebraically. $$ x^{2}+4 x=21 $$

Step-by-Step Solution

Answer

To solve the equation \(x^{2}+4x=21\), you need to first rewrite it into the standard form as \(x^{2}+4x-21=0\). Afterwards, use the quadratic formula to find the roots \(x_1\) and \(x_2\). Plot the function \(f(x)=x^{2}+4x-21\) to represent the solution graphically. \(x_1\) and \(x_2\) are where the curve intersects the x-axis. Finally, check the correctness of our solution by substituting \(x_1\) and \(x_2\) into our initial equation. If both sides match, then the solution is verified.

Step 1: Rewrite the Equation

Rewrite the given equation \(x^{2}+4x=21\) to standard form, which yields \(x^{2}+4x-21=0\)

Step 2: Solve the Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt {b^{2}-4ac}}{2a}\) to solve for \(x\). Inputs for the formula are \(a=1\), \(b=4\), and \(c=-21\). Calculating that yields two solutions \(x_1\) and \(x_2\).

Step 3: Graph the Function and Solutions

Plot the function \(f(x)=x^{2}+4x-21\). The solutions obtained will be the zeros of the function. The solutions \(x_1\) and \(x_2\) graphically represents the points where the curve intersects the x-axis.

Step 4: Check the Solution

Substitute the solutions \(x_1\) and \(x_2\) into the original equation to check the solution. If both sides of the equation are equal, then the solution is correct.

Key Concepts

Graphical Solution

A graphical solution involves plotting the quadratic function on a coordinate plane to visually identify its roots or solutions. In the exercise, you are given the equation \(x^2 + 4x = 21\). To represent it graphically, it first needs to be rewritten in the standard form, like this: \(x^2 + 4x - 21 = 0\). By visualizing this as \(f(x) = x^2 + 4x - 21\), you can create a graph.

To graph this function, find the vertex and sketch the parabolic curve. The solutions to the equation, also known as the zeros of the function, are found where the graph intersects the x-axis. These intersections, \(x_1\) and \(x_2\), are the points where \(f(x) = 0\), meaning the graph cuts the x-axis.

• Plot the vertex of the parabola first and determine the direction of the opening, which is upward in this case since the coefficient of \(x^2\) is positive.
• The intersections with the x-axis are your solutions.
• Verify these points by calculating their approximate x-values.

Using this graphical approach provides a visual understanding of how the function behaves and where its solutions lie.

Quadratic Formula

The quadratic formula is a reliable algebraic tool designed to find solutions to quadratic equations of the standard form \(ax^2 + bx + c = 0\). This formula is expressed as \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In our exercise, the quadratic equation is identified as \(x^2 + 4x - 21 = 0\), which means \(a = 1\), \(b = 4\), and \(c = -21\).

Here's how to apply the quadratic formula step-by-step:

• Identify coefficients from the equation: \(a = 1\), \(b = 4\), and \(c = -21\).
• Plug these values into the formula.
• Calculate the discriminant \((b^2 - 4ac)\) to determine the nature of solutions (real and distinct, real and repeated, or complex).
• Use the discriminant to find the values of \(x\).

The quadratic formula not only provides the exact solutions but also gives insight into the equation's characteristics based on the discriminant value. By calculating \(x_1\) and \(x_2\), you not only find the numerical solutions but also verify them against the graphical representation, offering a complete solution framework.

Zeros of a Function

Zeros of a function, often called the roots or solutions, are the x-values where the function evaluates to zero. For the quadratic function \(f(x) = x^2 + 4x - 21\), finding the zeros involves setting \(f(x) = 0\) and solving for \(x\).

These zeros are crucial because:

• They indicate where the graph of the function crosses or touches the x-axis.
• They demonstrate solutions of the quadratic equation when x-values that satisfy \(x^2 + 4x - 21 = 0\) are identified.
• They help understand the overall behavior and properties of the quadratic function.

For solving the zeros in our exercise, two primary approaches were discussed: using the quadratic formula and examining the graphical representation. Doing this algebraically confirms the precise locations of the zeros, while plotting the function on a graph offers an immediate visual verification of these solutions.

Understanding zeros is fundamental to solving quadratic equations. These solutions not only represent where the function zeroes out but also define key turning points and the symmetrical nature of the parabola in question. This dual algebraic and graphical confirmation ensures that the solutions are both accurate and well understood.