Problem 28 - Algebra for College Students

Question

Use the method of completing the square to solve each quadratic equation. $$x^{2}+5 x-3=0$$

Step-by-Step Solution

Answer

The solutions are \( x = -\frac{5}{2} \pm \frac{\sqrt{37}}{2} \).

Step 1: Move Constant to the Other Side

Start with the equation \( x^2 + 5x - 3 = 0 \). Move the constant term \(-3\) to the other side by adding 3 to both sides. The equation becomes \( x^2 + 5x = 3 \).

Step 2: Find the Number to Complete the Square

To complete the square, take the coefficient of \( x \), which is 5, divide it by 2 to get \( \frac{5}{2} \), and square it to get \( \left(\frac{5}{2}\right)^2 = \frac{25}{4} \).

Step 3: Add and Subtract the Square to Both Sides

Add \( \frac{25}{4} \) to both sides to maintain equality: \( x^2 + 5x + \frac{25}{4} = 3 + \frac{25}{4} \). This becomes \( x^2 + 5x + \frac{25}{4} = \frac{37}{4} \).

Step 4: Write the Quadratic as a Perfect Square

The left side of the equation can be written as a perfect square: \( \left(x + \frac{5}{2}\right)^2 \). So the equation becomes \( \left(x + \frac{5}{2}\right)^2 = \frac{37}{4} \).

Step 5: Solve for x by Taking the Square Root

Take the square root of both sides: \( x + \frac{5}{2} = \pm \sqrt{\frac{37}{4}} \). Simplifying, we get \( x + \frac{5}{2} = \pm \frac{\sqrt{37}}{2} \).

Step 6: Solve for x

Subtract \( \frac{5}{2} \) from both sides to solve for \( x \): \( x = -\frac{5}{2} \pm \frac{\sqrt{37}}{2} \). This gives the two solutions: \( x = -\frac{5}{2} + \frac{\sqrt{37}}{2} \) and \( x = -\frac{5}{2} - \frac{\sqrt{37}}{2} \).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions where the highest degree of the variable is squared (i.e., raised to the power of 2). These equations generally take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The standard quadratic equation often represents parabolic shapes when graphed.

Understanding quadratics can help in various fields such as physics, engineering, and economics as they can model real-world situations like projectile motion or optimization problems. Solving these equations enables us to find the points where the parabola intersects the x-axis, known as the roots or solutions of the equation.

When dealing with quadratic equations, there are multiple methods to find the solutions, such as factoring, using the quadratic formula, or completing the square. Each method offers a unique approach to uncovering the roots of the equation.

Solving Equations

Solving equations generally means finding the value of the variable that makes the equation true. In the context of quadratic equations, solving typically refers to finding the roots or the solutions where the equation equals zero.

One common method to solve such equations is completing the square, which involves turning the quadratic equation into a perfect square trinomial. This allows for simpler manipulation and resolution of the equation.

• Move the constant term to the other side of the equation.
• Determine the number needed to complete the square.
• Add and subtract this number on one side to keep the equation balanced.
• Rewrite the quadratic expression as a squared binomial.

Completing the square is particularly advantageous because it not just helps in solving the equation, but also in deriving the quadratic formula. It provides a systematic approach that emphasizes understanding over memorization.

Mathematical Methods

Mathematical methods refer to structured techniques used to solve mathematical problems, such as quadratic equations. One pivotal mathematical method in the realm of algebra is completing the square. This technique transforms a quadratic equation into a more manageable form, effectively unlocking the solution to the problem.

Completing the square breaks down into clear steps, allowing for precise manipulation of the mathematical expression:

• First, reposition terms to isolate the squared and linear terms of the equation.
• Deduce the perfect square by manipulating the middle coefficient.
• Transform the left-hand side into a squared term and adjust the right-hand side to maintain balance.
• The transformation results in a perfect square that can be easily solved by taking the square root.

This method enriches understanding by visually and algebraically revealing the symmetry and properties of the quadratic equation. As a mathematical tool, it equips students with a robust method to tackle quadratic equations beyond rote calculation.