Problem 30 - Basic Technical Mathematics with Calculus

Question

$$\text { Solve the given quadratic equations by factoring.}$$ $$2 x^{2}-7 a x+4 a^{2}=a^{2}$$

Step-by-Step Solution

Answer

The solutions are \(x = 2a\) and \(x = \frac{3a}{2}\).

Step 1: Set the equation to zero

First, we need to rewrite the equation so that all terms are on one side of the equation. Start by subtracting \(a^2\) from both sides of the equation: \[ 2x^2 - 7ax + 4a^2 - a^2 = 0 \] This simplifies to: \[ 2x^2 - 7ax + 3a^2 = 0 \]

Step 2: Identify and arrange the quadratic equation

The equation \( 2x^2 - 7ax + 3a^2 = 0 \) is now arranged as a standard quadratic equation \( ax^2 + bx + c = 0 \), where \( a = 2 \), \( b = -7a \), and \( c = 3a^2 \).

Step 3: Factor the quadratic equation

To factor \( 2x^2 - 7ax + 3a^2 \), we look for two numbers that multiply to \( 2 \times 3a^2 = 6a^2 \) and add to \(-7a\). The numbers \(-3a\) and \(-4a\) satisfy this condition (i.e., \(-3a \times -4a = 12a^2\) and \(-3a - 4a = -7a\)).

Step 4: Use the two numbers to split the middle term

Rewrite the equation as: \[ 2x^2 - 3ax - 4ax + 3a^2 = 0 \] Now we can factor by grouping. Group terms into pairs: \[ (2x^2 - 3ax) + (-4ax + 3a^2) = 0 \]

Step 5: Factor each group separately

Factor out common factors from each group: \[ x(2x - 3a) - a(4x - 3a) = 0 \] Notice that we can rewrite the expression so the factors are the same: \[ x(2x - 3a) - 2a(2x - 3a) = 0 \]

Step 6: Factor the common term

You can now factor out the common factor \((2x - 3a)\): \[ (x - 2a)(2x - 3a) = 0 \]

Step 7: Solve for x

Using the zero-product property, set each factor equal to zero: \(x - 2a = 0\) or \(2x - 3a = 0\). Solve each equation for \(x\):- \(x - 2a = 0\), so \(x = 2a\)- \(2x - 3a = 0\), so \(2x = 3a\) which simplifies to \(x = \frac{3a}{2}\)

Key Concepts

Factoring Quadratics

Factoring quadratics is a fundamental method used to solve quadratic equations. When you factor a quadratic, you essentially turn the quadratic expression into a product of two binomials. The original quadratic equation often appears in the form:

• \( ax^2 + bx + c = 0 \).

To factor it, we look for two numbers whose product equals the constant term \(a \times c\) and whose sum equals the linear coefficient \(b\). These two numbers help in splitting the middle term to facilitate grouping.

For the quadratic in the original exercise, we had to find two numbers that multiply to \(6a^2\) and add up to \(-7a\). It turns out that the numbers \(-3a\) and \(-4a\) fit the bill. By rewriting the quadratic expression with these two numbers, we can group terms and factor by grouping, resulting in the factors \((x - 2a)\) and \((2x - 3a)\).

Factoring is efficient for solving quadratics when the factors are integers or can be easily deduced, saving us the troubles of using more complex methods like the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation that might be tricky to factor outright. The formula itself looks like this:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

It is derived from completing the square on the standard form \( ax^2 + bx + c = 0 \). Here, \(b^2 - 4ac\) is known as the discriminant, and it tells us about the nature of the roots:

• A positive discriminant indicates two distinct real roots.
• A zero discriminant means there is one repeated real root.
• A negative discriminant suggests there are no real roots, only complex ones.

Even if our original problem was solved through factoring, the quadratic formula assures us that we have accounted for all possible roots, since it provides a precise solution for each kind. It is versatile and invaluable when factoring seems tough or if it leads to non-integer solutions.

Polynomial Equations

Polynomial equations are expressions consisting of variables raised to positive integer powers and their coefficients. Quadratic equations are a specific type of polynomial equation where the highest power of the variable is two, thus being known as second-degree polynomials.

• For instance, an equation like \(2x^2 - 7ax + 3a^2 = 0\) is a polynomial equation.

Solving polynomial equations like quadratics involves finding the values of the variable \(x\) that make the equation true. Methods such as factoring, using the quadratic formula, or graphing help us visualize or compute these points of intersection with the x-axis, known as roots or zeros of the polynomial.

While quadratics are manageable with straightforward tactics, higher-degree polynomials may require more complex approaches like synthetic division or numerical solutions. Understanding the particular characteristics of quadratic polynomials allows for efficient problem-solving, equipping students to tackle more advanced polynomial equations in the future.