Problem 22 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by completing the square. $$8 x^{2}+2 x=6$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3}{4}\) and \(x = -1\).

Step 1: Write the equation in standard form

First, make sure the quadratic equation is in the standard form. The given equation is \(8x^2 + 2x = 6\). Subtract 6 from both sides to get it to standard form: \(8x^2 + 2x - 6 = 0\).

Step 2: Simplify the equation

Divide the entire equation by 8 to simplify it: \(x^2 + \frac{1}{4}x - \frac{3}{4} = 0\). This makes it easier to complete the square.

Step 3: Isolate the quadratic and linear terms

Move the constant term to the other side by adding \(\frac{3}{4}\) to both sides of the equation: \(x^2 + \frac{1}{4}x = \frac{3}{4}\).

Step 4: Complete the square

To complete the square for the left-hand side expression \(x^2 + \frac{1}{4}x\), take half of the coefficient of \(x\), square it, and add it to both sides. Half of \(\frac{1}{4}\) is \(\frac{1}{8}\), and its square is \(\frac{1}{64}\). So, we add \(\frac{1}{64}\) to both sides: \(x^2 + \frac{1}{4}x + \frac{1}{64} = \frac{3}{4} + \frac{1}{64}\).

Step 5: Simplify both sides

The left side becomes a perfect square trinomial \((x + \frac{1}{8})^2\). For the right side, find the common denominator (64) to add fractions: \(\frac{3}{4} = \frac{48}{64}\), so \(\frac{3}{4} + \frac{1}{64} = \frac{49}{64}\). Therefore, the equation is \((x + \frac{1}{8})^2 = \frac{49}{64}\).

Step 6: Solve for x

Take the square root of both sides: \(x + \frac{1}{8} = \pm \frac{7}{8}\). This gives two equations after simplifying: \(x + \frac{1}{8} = \frac{7}{8}\) and \(x + \frac{1}{8} = -\frac{7}{8}\).

Step 7: Find the solutions

Solve the two equations: For \(x + \frac{1}{8} = \frac{7}{8}\), subtract \(\frac{1}{8}\) to get \(x = \frac{6}{8} = \frac{3}{4}\). For \(x + \frac{1}{8} = -\frac{7}{8}\), subtract \(\frac{1}{8}\) to get \(x = -\frac{8}{8} = -1\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra. They are polynomial equations of the second degree, typically expressed in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \) because if \( a \) were zero, the equation would no longer be quadratic but linear. Quadratics are essential because they often appear in diverse areas like physics, engineering, and economics, helping to model situations where a relationship between variables is not linear.

• Standard Form: This is \( ax^2 + bx + c = 0 \).
• Solutions: Quadratics can be solved using various methods—factoring, completing the square, or using the quadratic formula (\( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)). Completing the square is particularly useful for understanding the properties of quadratic functions, such as their vertex and direction.
• Graph: The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \( a \).

Recognizing a quadratic equation and knowing how to manipulate it provides a foundation for more advanced studies in mathematics.

Mathematics Problem Solving

Mathematics problem solving involves a logical approach to finding unknowns. It is about applying learned techniques to reach the solution effectively. Solving a quadratic equation like \( 8x^2 + 2x = 6 \) requires several steps to untangle its complexity.

• Identify the Form: First, rearrange the equation into its standard form. This enables a structured approach to manipulate the equation further.
• Simplification: This vital step often involves simplifying the equation by dividing all terms by the leading coefficient, making all subsequent steps more manageable.
• Strategic Approach: Choose the best method, like completing the square, based on the equation type and your familiarity with techniques. This strategy-oriented mindset assures that you narrow in on an efficient path to the solution.

Problem solving in math is not just about the answer; it’s about understanding the journey. Each step is crucial towards peeling back layers of the problem to uncover the unknown.

Algebraic Techniques

Algebraic techniques form the backbone of solving quadratic equations through methods like completing the square. It involves manipulating algebraic expressions smartly.

• Completing the Square: This technique turns a quadratic expression into a perfect square trinomial. For example, with \( x^2 + \frac{1}{4}x \), you add and subtract the square of half the coefficient of \( x \). This method not only aids in finding solutions but also helps understand the properties of quadratic graphs.
• Isolation of Terms: Key to solving quadratic equations, this involves moving terms to one side to simplify the process of completing the square.
• Simplifying & Solving: By completing the square, the quadratic equation becomes easier to handle. You factor it into a square term such as \((x + p)^2 = q\), which can then be solved by taking the square root, leading to solving for \( x \).

Algebraic techniques aren’t just about performing operations; they offer a systematic approach to dissecting and solving complex mathematical problems efficiently.