Problem 24 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \(x^{2}-3 x-5=0\)

Step-by-Step Solution

Answer

The solutions of the equation are \(x=1.5 + \sqrt{7.25}\) and \(x=1.5 - \sqrt{7.25}\).

Step 1: Rewrite the equation

To solve the quadratic equation \(x^{2}-3 x-5=0\) by completing the square, rewrite the equation with the constant term on the right side of the equation. The equation becomes \(x^{2}-3 x=5\).

Step 2: Calculate the square completion

Next, a term needs to be added to complete the square. This term can be found by taking half of the coefficient of the linear term, squaring it, and then adding it to both sides of the equation. The coefficient of the linear term is -3. Half of -3 is -1.5, and \((-1.5)^2 = 2.25\). Thus, the modified equation is \(x^{2}-3x + 2.25=5 + 2.25\), simplified to \(x^{2}-3x + 2.25=7.25\).

Step 3: Rewrite into perfect square

We can now rewrite the left side of the equation as a perfect square. The left side becomes \((x-1.5)^2\), and the equation is \((x-1.5)^2 =7.25\).

Step 4: Solve for x

Finally, to solve for x, take the square root of both sides. Remember to consider both the positive and negative square roots. Thus, \(x-1.5 =\pm \sqrt{7.25}\). Therefore, \(x=1.5 \pm \sqrt{7.25}\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. This type of equation is called "quadratic" because the highest power of the variable \(x\) is 2, which makes it a second-degree polynomial equation. When you encounter a quadratic equation, the goal is to find the value(s) of \(x\) that make the equation true.

A quadratic equation can appear in many forms. The equation in this example is written as \(x^2 - 3x - 5 = 0\). To solve a quadratic equation, you often need to rearrange or transform it to make it easier to find the solution. One popular method for solving quadratic equations is completing the square, which helps turn any quadratic equation into a form that is easy to solve.

Creating a Perfect Square

Completing the square involves manipulating a quadratic equation to form a "perfect square trinomial." A perfect square trinomial is a quadratic expression that can be written as \((x - p)^2\), where \(p\) is some number.

To complete the square for an equation, you first need to look at the quadratic and linear terms. Consider the equation \(x^2 - 3x\). You take half of the coefficient of the linear term, which is \(-3\). Half of \(-3\) is \(-1.5\). Then, you square \(-1.5\) to get \(2.25\). By adding \(2.25\) to both sides of the equation, you create a perfect square trinomial on the left.

Once the left side is a perfect square trinomial, it can be factored neatly into \((x - 1.5)^2\). This form makes it straightforward to solve the equation since it's much easier to work with squares.

Steps to Solving Equations Using the Square

After converting the left side of a quadratic equation into a perfect square, the next step is solving the equation. Our goal is to isolate \(x\) so we need to take the square root of both sides. This step requires considering both the positive and negative square roots.

For the equation \((x - 1.5)^2 = 7.25\), take the square root of both sides to get \(x - 1.5 = \pm \sqrt{7.25}\). This means \(x\) can be \(1.5 + \sqrt{7.25}\) or \(1.5 - \sqrt{7.25}\). To present these as solutions, calculate the approximate values of the square roots to find the numerical values of \(x\).

• Add \(1.5\) to the positive and negative roots of \(7.25\) to find the two solutions.
• Remember that completing the square is a method that works for any quadratic equation, not just the ones that are easy to factor or solve using simpler methods.