Problem 26 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$4 x^{2}+25=20 x$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{5}{2}\).

Step 1: Move All Terms to One Side

First, we need to rewrite the equation so that all terms are on one side and zero is on the other side. Initially, the equation is \(4x^2 + 25 = 20x\). Subtract \(20x\) from both sides, resulting in \(4x^2 - 20x + 25 = 0\).

Step 2: Recognize the Structure

Notice that the quadratic equation we have is \(4x^2 - 20x + 25 = 0\). It looks like it might be a perfect square trinomial, which typically has the form \((ax + b)^2\).

Step 3: Factor the Quadratic

Identify the factors by grouping or using the standard factoring approach. Our equation \(4x^2 - 20x + 25\) can be rewritten as \((2x - 5)^2 = 0\). Check this by expanding \((2x - 5)^2\): \((2x - 5)(2x - 5) = 4x^2 - 10x - 10x + 25 = 4x^2 - 20x + 25\).

Step 4: Solve for the Roots

Now that we have factored the equation as \((2x - 5)^2 = 0\), we solve for \(x\) by setting the factor equal to zero. Thus, \(2x - 5 = 0\).

Step 5: Finalize the Solution

Solve for \(x\) by isolating it: \(2x = 5\). Divide both sides by 2 to get \(x = \frac{5}{2}\). This is the only solution as it is a repeated root.

Key Concepts

Factoring

Factoring is a fundamental technique in algebra used to simplify or solve polynomial equations, particularly quadratic equations. A quadratic equation is generally expressed in the form \(ax^2 + bx + c = 0\). Factoring means breaking down this equation into simpler expressions, or "factors," that, when multiplied together, give back the original quadratic equation. This method simplifies the process of finding solutions or "roots," the values of \(x\) that satisfy the equation.

Consider the equation \(4x^2 - 20x + 25 = 0\). The goal of factoring is to express this equation as a product of simpler factors. By using factoring techniques, often by recognizing patterns or using trial and error strategies, we try to decompose this equation into two binomial expressions, such as \((dx - e)(fx - g)\). Once factored, it's easier to solve the equation by setting each factor equal to zero and solving for \(x\).

When a quadratic equation is factored, solving the equation becomes an exercise in simple arithmetic. Factoring is efficient and particularly useful when the quadratic expression is easily recognizable as a "special form," like a perfect square trinomial.

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic expression that takes the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\). Recognizing this form is useful because it simplifies the process of factoring, as the trinomial is already structured in a way that suggests its factors.

In the exercise, the expression \(4x^2 - 20x + 25\) is identified as a perfect square trinomial, because it can be rewritten as \((2x - 5)^2\):

• The coefficient of \(x^2\), which is 4, is \((2)^2\).
• The constant term, 25, is \((-5)^2\).
• The middle term, -20x, is twice the product of \(2\) and \(-5\) (i.e., \(2 \cdot 2 \cdot -5 = -20\)).

Having identified these components, we can confirm that \(4x^2 - 20x + 25\) fits the perfect square trinomial form. This helps us factor it effortlessly into \((2x - 5)^2\). Recognizing and utilizing perfect square trinomials makes finding solutions more straightforward and highlights the elegance of algebraic manipulation.

Roots of an Equation

The roots of a quadratic equation are the values of \(x\) that satisfy the equation. These are the points where the quadratic function crosses the \(x\)-axis on a graph. In many cases, a quadratic equation in the standard form \(ax^2 + bx + c = 0\) will have two distinct solutions, two repeated (or identical) solutions, or no real solutions.

For this particular quadratic equation, after factoring it into \((2x - 5)^2 = 0\), we solve for the roots by setting the factor equal to zero:

• \(2x - 5 = 0\)

Solve the linear equation by isolating \(x\) to get:

• \(2x = 5\)
• \(x = \frac{5}{2}\)

This results in a repeated root, indicating that the parabola represented by the quadratic equation touches the \(x\)-axis at only one point, \(x = \frac{5}{2}\). Repeated roots occur in cases of perfect square trinomials and imply that the graph of the quadratic "just kisses" the \(x\)-axis, rather than cutting through it.