Problem 21 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$x^{2}=4 x$$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}=4x\) are \(x = 0\) and \(x = 4\).

Step 1: Simplify the given equation

The given equation is \(x^{2}=4x\). It can be rearranged and simplified into a proper quadratic form by subtracting \(4x\) from both sides, which yields \(x^{2} - 4x = 0\)

Step 2: Factor the equation

To factor the equation \(x^{2} - 4x = 0\), observe that both terms have a common factor of \(x\). Thus, the factored form of the equation will be \(x(x - 4) = 0\)

Step 3: Solve for x

To find the solutions, set each factor equal to zero and solve for \(x\). Therefore, \(x= 0\) and \(x - 4 = 0 \Rightarrow x = 4\). Hence, the roots of the equation are \(x = 0\) and \(x = 4\)

Step 4: Check by substitution or graphing

Substitute both values of \(x\) into the original equation and check if both left side and right side of the equation are equal. When \(x = 0\), \(x^{2} = 4x \Rightarrow 0 = 0\) and when \(x = 4\), \(x^{2} = 4x \Rightarrow 16 = 16\). Alternatively, the check can be done by graphing the equation \(x^{2}=4x\). When x=0 and x=4, the graph should intersect x-axis, meaning those points are x-intercepts of the graph.

Key Concepts

Factoring Quadratic Equations

Factoring is a valuable skill when solving quadratic equations, as it simplifies the process of finding the roots of the equation. To factor a quadratic equation, identify the greatest common factor (GCF) of the terms. For the equation \(x^{2} = 4x\), rearranging gives us \(x^{2} - 4x = 0\), revealing that \(x\) is a common factor.

Factor out the GCF to write the quadratic in the form \(x(a - b) = 0\), where \(a\) and \(b\) are the remaining terms. In our example, this yields \(x(x - 4) = 0\). Since a product equals zero if and only if one of its factors is zero, set each factor equal to zero: \(x = 0\) and \(x - 4 = 0\), providing us with the roots of the equation.

Quadratic X-Intercepts

The x-intercepts of a graph, also known as 'roots' or 'zeros', are points where the graph crosses the x-axis. In the context of quadratic equations, the x-intercepts can be found by solving the equation \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.

In our example \(x^{2} - 4x = 0\), setting the factored form \(x(x - 4) = 0\) equal to zero leads to two x-intercepts: \(x = 0\) and \(x = 4\). These intercepts represent the points at which the parabola defined by the equation will touch or cross the x-axis. Detecting x-intercepts allows us to visualize the graph and understand the behavior of the quadratic function.

Graphing Quadratic Equations

Graphing is a powerful tool to visualize the behavior of quadratic equations, which typically form a U-shaped curve called a parabola. When graphing \(x^2 = 4x\), or in its equivalent factored form \(x(x - 4) = 0\), we identify the x-intercepts (0 and 4) where the parabola meets the x-axis.

The vertex, or the highest or lowest point of the parabola, is central to understanding the graph's shape. For the equation \(x^2 - 4x\), the vertex lies at the midpoint between the x-intercepts at \(x = 2\). By plotting a few more points, we can sketch the parabola opening upwards since the coefficient of \(x^2\) is positive and passing through the identified intercepts. This visual representation not only confirms the solutions but also provides insight into the equation's maximum and minimum values, as well as its symmetry.