Problem 27 - Elementary Algebra

Question

Solve each quadratic equation by completing the square. $$ 2 x^{2}+7 x-12=0 $$

Step-by-Step Solution

Answer

Answer: The solutions are \(x = -\frac{7}{4} + \sqrt{\frac{145}{16}}\) and \(x = -\frac{7}{4} - \sqrt{\frac{145}{16}}\).

Step 1: Rewrite the equation in the standard form

The given equation is already in the standard form: $$ 2x^2+7x-12=0 $$

Step 2: Divide the equation by the coefficient of the x^2 term

Divide the entire equation by 2: $$ x^2+\frac{7}{2}x-6=0 $$

Step 3: Complete the square

To make a perfect square trinomial, we need to add the square of half of the coefficient of the x term: $$ \left(\frac{\frac{7}{2}}{2}\right)^2 = \left(\frac{7}{4}\right)^2 = \frac{49}{16} $$ Add \(\frac{49}{16}\) to both sides of the equation: $$ x^2 + \frac{7}{2}x + \frac{49}{16} = 6 + \frac{49}{16} $$

Step 4: Rewrite the left side of the equation as a binomial squared

Now, we rewrite the left side of the equation as a binomial squared: $$ \left(x+\frac{7}{4}\right)^2 = 6 + \frac{49}{16} $$

Step 5: Solve for x using square roots

Solve for x by taking the square root of both sides of the equation: $$ x + \frac{7}{4} = \pm\sqrt{6 + \frac{49}{16}} $$ Now, simplify the right side of the equation: $$ x + \frac{7}{4} = \pm\sqrt{\frac{96 + 49}{16}} = \pm\sqrt{\frac{145}{16}} $$ Finally, subtract \(\frac{7}{4}\) from both sides to solve for x: $$ x = -\frac{7}{4} \pm\sqrt{\frac{145}{16}} $$ The two solutions for x are: $$ x = -\frac{7}{4} + \sqrt{\frac{145}{16}},\, -\frac{7}{4} - \sqrt{\frac{145}{16}} $$

Key Concepts

Quadratic Equations

To begin solving any quadratic equation by completing the square, it's essential to understand what a quadratic equation is. At its core, a quadratic equation is any equation that forms a parabola when graphed. It's typically written in the standard form of:
\[ ax^2 + bx + c = 0 \]
where:

• \(a\), \(b\), and \(c\) are constants, with \(a\) not equal to zero.
• \(x\) represents the unknown variable.

Quadratic equations can have two solutions, one solution, or no real solutions at all, depending on the discriminant \(b^2 - 4ac\). Understanding this foundation will help you dive into methods like completing the square, which is one of the most systematic ways to find the roots of quadratic equations.

Binomial Squared

A binomial squared is a perfect square trinomial that can be expressed in the form:
\[ (x + d)^2 \]
For example, \((x + 3)^2\) results in \(x^2 + 6x + 9\).
This technique is crucial in the process of completing the square because it allows us to simplify complex expressions into a format that is easy to solve using square roots.
When completing the square, you're transforming part of your quadratic equation into this structure. You do this by adding and subtracting the square of half the coefficient of the middle term \(b\) — it’s this manipulation that allows the expression to neatly fold into the squared form.
Building the intuition for recognizing a binomial squared helps streamline the solving process and sets the stage for finding solutions more easily.

Solving Quadratic Equations

Solving quadratic equations by completing the square involves transforming your quadratic expression into an equation featuring a perfect square, making it possible to find the roots by taking square roots. Here’s a quick rundown of the steps you would ordinarily follow:

• First, ensure the quadratic equation is in the standard form.
• Divide the equation by the coefficient of \(x^2\) (if it’s not 1) to simplify it, thus preparing it for completing the square.
• Transform the rearranged quadratic into a binomial squared by adding/subtracting the necessary constant.
• Once in perfect square form, solve for \(x\) by taking the square root of both sides.

These steps create a clear path through the complexity of the quadratic equation, allowing you to isolate and solve for \(x\) straightforwardly. This method is particularly useful for equations that do not factor easily.

Algebraic Manipulation

Algebraic manipulation is all about rearranging and simplifying expressions to achieve the desired form or solution. It’s an invaluable skill when solving quadratic equations by completing the square.
Let’s break down what it involves:

• Rewriting: You might need to rewrite terms, such as separating or combining like terms or transforming expressions into equivalent forms.
• Balancing: When you add or subtract a number to one side of an equation, ensure you do the same operation to the other side to maintain equality.
• Simplifying: By completing the square, you simplify the quadratic expression to a format that essentially handles the variable manipulation for you.

Think of algebraic manipulation as the toolset for crafting your quadratic equation into a solvable format. It involves a mix of techniques like factoring, distributing, and completing the square itself, each guiding you towards the quadratic equation’s solutions.