Problem 16 - Algebra for College Students

Question

Use the method of completing the square to solve each quadratic equation. $$x^{2}+2 x-1=0$$

Step-by-Step Solution

Answer

The solutions are \(x = -1 + \sqrt{2}\) and \(x = -1 - \sqrt{2}\).

Step 1: Move Constant to the Other Side

First, we need to rearrange the equation so that the constant term is on the right side. Start with the given equation:\[ x^2 + 2x - 1 = 0 \]Add 1 to both sides to get:\[ x^2 + 2x = 1 \]

Step 2: Identify the Coefficient for Completing the Square

To complete the square, we need to focus on the coefficient of the linear term \(2x\). Identify the coefficient of \(x\) and divide it by 2:\[ \frac{2}{2} = 1 \]

Step 3: Square the Result

Take the result from the previous step and square it:\[ (1)^2 = 1 \]This value, 1, will be added to both sides of the equation to complete the square.

Step 4: Add the Squared Result to Both Sides

Now, add 1 to both sides of the equation:\[ x^2 + 2x + 1 = 1 + 1 \]This simplifies to:\[ x^2 + 2x + 1 = 2 \]

Step 5: Rewrite the Left Side as a Perfect Square

The left side of the equation \(x^2 + 2x + 1\) can now be written as a perfect square:\[ (x + 1)^2 = 2 \]

Step 6: Solve for x by Taking the Square Root

Take the square root of both sides of the equation:\[ x + 1 = \pm \sqrt{2} \]

Step 7: Solve for x

Finally, solve for \(x\) by subtracting 1 from both sides:\[ x = -1 \pm \sqrt{2} \]

Key Concepts

Understanding Quadratic Equations

Quadratic equations are an essential part of algebra and appear in the form of \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants. The variable \(x\) is what we solve for. In our exercise, the equation \(x^2 + 2x - 1 = 0\) follows this standard format with \(a = 1\), \(b = 2\), and \(c = -1\). Quadratics can visually represent parabolas on a graph and solving these equations gives the points where the curve intersects the x-axis. This makes quadratic equations highly valuable in physics, engineering, and various areas where modeling curved paths is necessary. Quadratics can often be solved by factoring, using the quadratic formula, or by completing the square—a method we focus on in this exercise.

The Technique of Solving Equations by Completing the Square

Completing the square is a method used to solve quadratic equations by transforming them into perfect square trinomials. This technique involves a few clear steps:

• Rearrange: Start by moving the constant term to the other side of the equation. For \(x^2 + 2x - 1 = 0\), add 1 to both sides, which gives \(x^2 + 2x = 1\).
• Coefficient Analysis: Look at the coefficient of the \(x\) term (here it is 2), divide by 2, then square it (\( (\frac{2}{2})^2 = 1\)). This gives the number we'll add to both sides of the equation.
• Construct the Square: The equation now becomes \(x^2 + 2x + 1 = 2\), due to adding 1 to both sides, which makes the left side a perfect square (\((x + 1)^2\)).

Through these steps, the equation transforms for easier solving in the next stage. Completing the square is not only useful for solving quadratics but also in deriving the quadratic formula itself and helping in calculus for integrating functions.

Using Square Roots in Solving Quadratic Equations

Taking the square root is a critical step in solving equations, especially after completing the square. Consider

• Once we have \((x + 1)^2 = 2\), we take the square root of both sides to simplify. This gives \(x + 1 = \pm \sqrt{2}\). Note the \(\pm\) symbol, indicating two potential solutions because both positive and negative roots of real numbers can solve the original equation.
• Solve for \(x\) by isolating it. Subtract 1 from \(x + 1 = \pm \sqrt{2}\) to get \(x = -1 \pm \sqrt{2}\). This step solves the quadratic equation, revealing that \(x\) can be either \(-1 + \sqrt{2}\) or \(-1 - \sqrt{2}\).

Square roots are thus vital in finding exact roots of quadratic equations and understanding which points intersect on a graph. This methodical and accurate approach is critical in mathematics, helping us predict and analyze real-world scenarios efficiently.