Problem 34 - Elementary Algebra

Question

For the following problems, solve the equations by completing the square. $$ 7 a^{2}+3 a-1=0 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation $$7a^2 + 3a - 1=0$$ by completing the square method, and find the possible values of $$a$$. Answer: The solutions for $$a$$ are $$a = \frac{3 + \sqrt{33}}{14}$$ or $$a = \frac{3 - \sqrt{33}}{14}$$.

Step 1: Make the coefficient of $$a^2$$ term equal to 1

Divide the equation by the coefficient of $$a^2$$ (which is 7) to make it 1: $$ \frac{7a^2}{7} + \frac{3a}{7} - \frac{1}{7} = 0 $$ Simplifying, we get $$ a^2 + \frac{3}{7} a - \frac{1}{7} = 0 $$

Step 2: Add and subtract the square of half of the coefficient of $$a$$ term

Take half of the coefficient of $$a$$ term ($$\frac{3}{7}$$), square it, and add and subtract that value inside the parenthesis. We get, $$ a^2 + \frac{3}{7} a + \left(\frac{3}{14}\right)^2 - \left(\frac{3}{14}\right)^2 - \frac{1}{7} = 0 $$

Step 3: Rewrite the equation as a perfect square trinomial

The first three terms make a perfect square trinomial. Write the equation as $$ \left(a + \frac{3}{14}\right)^2 - \frac{9}{196} - \frac{1}{7} = 0 $$

Step 4: Solve for $$a$$ by taking square roots of both sides

Combine the constants and move them to the other side of the equation, then take the square root of both sides: $$ \left(a + \frac{3}{14}\right)^2 = \frac{9}{196} + \frac{1}{7} $$ Solve for $$a$$: $$ a + \frac{3}{14} = \pm \sqrt{\frac{9}{196} + \frac{1}{7}} $$ Subtract the constant term: $$ a = \frac{3}{14} \pm \sqrt{\frac{9}{196} + \frac{1}{7}} $$ Simplifying further, we get the two possible values of $$a$$: $$ a = \frac{3 + \sqrt{33}}{14} \quad \text{or} \quad a = \frac{3 - \sqrt{33}}{14} $$

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of a single variable with a degree of two. In other words, the highest power of the variable is 2. These equations generally take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are constants, and \(a\) is not zero. Solving such equations often involves finding the values of the variable, commonly denoted as \(x\), which satisfy the equation. These solutions are referred to as the "roots" of the equation.
To solve quadratic equations, various methods can be used, such as factoring, using the quadratic formula, or completing the square. Each method has its own set of steps and is suitable for different types of equations. Completing the square, for instance, is particularly useful when the quadratic equation is not easily factorable.

Algebraic Solutions

Algebraic solutions refer to finding the roots of an equation using algebraic manipulations and logical steps. In the case of quadratic equations, algebraic solutions could involve:

• Factorization: Expressing the quadratic equation as a product of two binomials. This method is efficient but depends on whether the quadratic can be easily factored.
• Quadratic Formula: A formula that provides the solutions directly as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the Square: Rewriting the equation in the form \((x + d)^2 = e\), which is more systematic for any quadratic equation.

Completing the square involves adjusting the equation to make it a perfect square trinomial, which then allows the equation to be transformed into a simpler form that's easier to solve. By doing so, algebraic solutions enable one to understand the roots in a structured manner.

Perfect Square Trinomials

A perfect square trinomial is a particular kind of polynomial that results from squaring a binomial. It can be represented in the standard form \((x + y)^2 = x^2 + 2xy + y^2\). The notable feature of a perfect square trinomial is that it can easily be rewritten as a square of a binomial.
To create a perfect square trinomial during the process of solving quadratic equations, you take the following steps:

• Identify the coefficient of the linear term (\(b\) in \(ax^2 + bx + c\)).
• Divide it by 2 to find \(y\).
• Square it and add or subtract this value to complete the square.

This process is key to transforming the quadratic equation into a form that can be easily solved. In our example, adding and subtracting \(\left(\frac{3}{14}\right)^2\) allowed the expression to form a perfect square trinomial. Once rewritten, it simplifies the calculation of the variable \(a\) by enabling the use of the square root operation.