Problem 18 - Algebra 2 and Trigonometry

Question

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ x^{2}+2 x-5=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = -1 + \sqrt{6} \) and \( x = -1 - \sqrt{6} \).

Step 1: Rearrange the Equation

First, we will start with the quadratic equation given: \( x^2 + 2x - 5 = 0 \). In completing the square, it is essential to have the constant term on the right-hand side. So we rewrite the equation as follows: \( x^2 + 2x = 5 \).

Step 2: Identify and Add the Square Term

To complete the square, we take the coefficient of \( x \) which is \( 2 \), divide it by \( 2 \), and then square the result. This gives \( (2/2)^2 = 1 \). We add this term inside the equation, ensuring to add the same to the right side of the equation too. Thus, we have: \( x^2 + 2x + 1 = 5 + 1 \).

Step 3: Form a Perfect Square Trinomial

Now, the left side of the equation \( x^2 + 2x + 1 \) is a perfect square trinomial and can be expressed as \( (x+1)^2 \). Thus, the equation becomes \( (x+1)^2 = 6 \).

Step 4: Solve for x

To solve for \( x \), take the square root of both sides: \( \sqrt{(x+1)^2} = \sqrt{6} \). This simplifies to \( x + 1 = \pm \sqrt{6} \). Finally, subtract \( 1 \) from both sides to isolate \( x \): \( x = -1 \pm \sqrt{6} \).

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of degree 2. It is generally written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).
One significant feature of quadratic equations is their parabolic graph, which can open upwards or downwards depending on the sign of \( a \). Solving quadratic equations can be done by various methods, including:

• Factoring
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square

Completing the square simplifies the equation into a format that reveals the solutions directly and is particularly useful when other methods are not feasible.
For the exercise \( x^2 + 2x - 5 = 0 \), completing the square is an effective approach to express solutions in the simplest radical form.

Simplest Radical Form

When you solve quadratic equations by taking square roots, you often end up with solutions involving square roots. Expressing these solutions in simplest radical form means simplifying the expression until no further simplification is possible.
This involves:

• Ensuring the radicand (the number inside the radical) has no perfect square factors other than 1.
• Rationalizing denominators, if any, to eliminate radicals.

For instance, the solution \( x = -1 \pm \sqrt{6} \) is in simplest radical form because \( 6 \) does not have a perfect square factor other than 1, and the expression does not require further simplification. This form is beneficial for providing precise answers without decimal approximations.

Perfect Square Trinomial

A perfect square trinomial is an algebraic expression that can be factored into a squared binomial. It takes the form \( (a+b)^2 = a^2 + 2ab + b^2 \), which allows it to be rewritten more concisely as \( a^2 + 2ab + b^2 \).
Recognizing or creating a perfect square trinomial is a key step in completing the square.
To form a perfect square trinomial:

• Take half of the linear coefficient (the \( b \) in \( ax^2 + bx + c \)).
• Square this value to find the constant term to add or adjust in the equation.

In the example provided, \( x^2 + 2x + 1 \) is transformed by completing the square into \((x+1)^2\). This demonstrates how the quadratic can be rewritten as a perfect square trinomial, simplifying the problem-solving process.