Problem 26 - Algebra 2 and Trigonometry

Question

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ \frac{1}{4} x^{2}+\frac{3}{4} x-\frac{3}{2}=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{3}{2} + \frac{\sqrt{33}}{2}\) and \(x = -\frac{3}{2} - \frac{\sqrt{33}}{2}\).

Step 1: Clear the Fraction Coefficients

To make the equation easier to work with, eliminate fractions by multiplying every term by 4. This gives:\[x^2 + 3x - 6 = 0\]

Step 2: Move Constant Term

Move the constant term to the other side of the equation:\[x^2 + 3x = 6\]

Step 3: Complete the Square

To complete the square, take half of the coefficient of the linear term (\(3\)), square it, and add it to both sides. Half of 3 is \(\frac{3}{2}\), and squaring it gives \(\frac{9}{4}\). Add this to both sides:\[x^2 + 3x + \frac{9}{4} = 6 + \frac{9}{4}\]

Step 4: Simplify the Right Side

Convert 6 to a fraction with a denominator of 4 to add \(\frac{9}{4}\).\[6 = \frac{24}{4}, \quad \frac{24}{4} + \frac{9}{4} = \frac{33}{4}\]Now the equation is:\[(x + \frac{3}{2})^2 = \frac{33}{4}\]

Step 5: Solve for x

Take the square root of both sides:\[x + \frac{3}{2} = \pm \sqrt{\frac{33}{4}}\]Simplify the right side using the property \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\):\[x + \frac{3}{2} = \pm \frac{\sqrt{33}}{2}\]Then, solve for \(x\):\[x = -\frac{3}{2} \pm \frac{\sqrt{33}}{2}\]

Step 6: Express in Simplest Radical Form

Express the two solutions separately:\[x = -\frac{3}{2} + \frac{\sqrt{33}}{2}\]\[x = -\frac{3}{2} - \frac{\sqrt{33}}{2}\]

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation where the highest degree of the variable is two. It generally takes the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The equation we're working with is a bit unique since it originally contains fractions: \( \frac{1}{4}x^2 + \frac{3}{4}x - \frac{3}{2} = 0 \). By multiplying each term by 4, we eliminate the fractions, yielding \( x^2 + 3x - 6 = 0 \). This step simplifies the problem significantly, making subsequent operations simpler, particularly for solving by completing the square.

Simplest Radical Form

Expressing a solution in its simplest radical form involves simplifying the square roots and fractions as much as possible. In our example, after completing the square, we derive \((x + \frac{3}{2})^2 = \frac{33}{4}\). Taking the square root of both sides, we get \(x + \frac{3}{2} = \pm \frac{\sqrt{33}}{2}\). Here, \(\sqrt{33}\) is the simplest radical form because 33 can't be reduced further since it isn't a perfect square and isn't divisible by any perfect squares other than 1. The division simplifies the expression and helps in clearly understanding the solutions' values.

Solving Quadratic Equations

There are several methods to solve quadratic equations, and one of the most precise is completing the square. This method involves rearranging the equation so that one side forms a perfect square trinomial. In our case, after moving the constant to one side and finding an appropriate value to complete the square (\(\frac{9}{4}\)), we convert the left-hand side to \((x + \frac{3}{2})^2\). The beauty of completing the square lies in transforming equations into forms that are straightforward to solve, often providing solutions quickly. Once expressed as a square, solving for \(x\) by taking the square root of both sides becomes trivial, finally leading to the simplified radical form solutions: \(x = -\frac{3}{2} \pm \frac{\sqrt{33}}{2}\). This method is a reliable step-by-step approach to handle quadratics that may otherwise seem challenging to solve directly.