Problem 58 - College Algebra

Question

Solve each equation in Exercises \(47-64\) by completing the square. $$x^{2}-3 x-5=0$$

Step-by-Step Solution

Answer

The solutions for the equation are \(x = 4.19\) and \(x = -1.19\).

Step 1: Move the Constant to the Right Side of the Equation

Firstly, isolate the x squared and x terms on one side of the equation by moving the constant on the other side of the equation. This gives \(x^{2} - 3x = 5\).

Step 2: Complete the Square

The middle term of our quadratic equation in the 'x' is -3. We halve this coefficient, and square it. \((-3/2)^2 = 2.25\). Add this to both sides of the equation to get a perfect square trinomial on the left. This means modifying our equation to \(x^{2} - 3x + 2.25 = 5 + 2.25\), simplifying to \( (x-1.5)^2 = 7.25\). We took the square root of 2.25, which is 1.5, to decide what (x - 1.5) should be.

Step 3: Solve for 'x'

Now that we have a perfect square on one side, we can solve for 'x'. To get 'x' alone, we take the square root of both sides of the equation and then add 1.5 to both sides, hence: \(x = 1.5 ± \sqrt{7.25}\), which simplifies to \(x = 1.5 ± 2.69\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree. It means this equation has the form:

• \(ax^2 + bx + c = 0\)

Here, \(x\) represents an unknown, while \(a\), \(b\), and \(c\) are constants with \(a eq 0\). This ensures our equation's degree remains quadratic. Quadratic equations can have two solutions, which are the values of \(x\) for which the equation becomes zero.

Solving quadratic equations can be done using various methods such as factoring, using the quadratic formula, or completing the square.

• Factoring involves writing the quadratic expression as a product of two binomials and setting each binomial to zero to find the value of \(x\).
• Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) provides a straightforward way of finding solutions.
• Completing the square is a method involving transforming the equation into a perfect square trinomial on one side, which can be easily solved.

Among these, completing the square can help deepen understanding by visually demonstrating the balance and symmetry in an equation.

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic expression. It looks like this:

• \((x + b)^2 = x^2 + 2bx + b^2\)

It is the result of squaring a binomial. The equation \(x^2 - 3x + 2.25\) in the solution is a perfect square trinomial. Here, the coefficient \(-3\) is halved to \(-1.5\), and squaring this provides \(2.25\), forming the complete trinomial.

This transformation helps convert a more complicate form into something manageable, making it easier to solve for \(x\). It's a crucial step in the process of completing the square.
The benefit of a perfect square trinomial is the simplification it brings:

• You can recognize it quickly.
• It represents a perfect square, meaning it can be expressed as \((x + b)^2\), effectively reducing the degree of solving the quadratic.

Understanding how to form and utilize a perfect square trinomial is an invaluable tool in algebra.

Square Root

The square root function is essential in solving quadratic equations, especially after forming a perfect square. When you have a perfect square like \((x - 1.5)^2 = 7.25\), taking the square root on both sides is a logical step to isolate \(x\).

Here’s how it works:

• Apply the square root to both sides: \(\sqrt{(x - 1.5)^2} = \sqrt{7.25}\).
• This simplifies to \(x - 1.5 = \pm \sqrt{7.25}\).
• Finally, solve for \(x\) by adding \(1.5\) to both sides to find the solutions: \(x = 1.5 \pm 2.69\).

The square root first gives us two possible solutions because \(\sqrt{a^2} = \pm a\). It reflects the natural property of square roots having two outcomes: one positive and one negative.

Learning to effectively handle square roots within algebraic expressions is key to mastering quadratic equations and their solutions.