Problem 32 - Introductory Algebra for College Students

Question

Solve each quadratic equation by first factoring the perfect square trinomial on the left side. Then apply the square root property. Simplify radicals, if possible. $$x^{2}+4 x+4=25$$

Step-by-Step Solution

Answer

The solutions for the equation \(x^{2}+4 x+4=25\) are \(x = 3\) and \(x = -7\)

Step 1: Identifying and Factoring the Perfect Square Trinomial

The given equation, \(x^{2}+4 x+4=25\), can be identified as a perfect square trinomial. The perfect square trinomial is \((x+2)^{2}\). Thus, the equation can be rewritten as: \((x+2)^{2}=25\)

Step 2: Applying the Square Root Property

Now, we will apply the square root property to the equation obtained from the previous step, \((x+2)^{2}=25\). Here, the square root property can be applied by taking the square root of both sides of the equation. This yields: \(x + 2 = \pm \sqrt{25}\)

Step 3: Final Calculation and Simplification

Solving from the last step, we have \(x+2= \pm 5\). Subtracting 2 from both sides gives us possible solutions for x: \(x = 5 - 2 = 3\) when considering the positive root, and \(x = -5 - 2 = -7\) when considering the negative root

Step 4: Checking the Solutions

Substitute the solutions (x = 3 and x = -7) back into the original equation (\(x^{2}+4 x+4=25\)) to ensure they hold true

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression. It has the form

• \( (a + b)^2 = a^2 + 2ab + b^2 \)

where both \(a^2\) and \(b^2\) are perfect squares. The given quadratic equation, \(x^2 + 4x + 4\), is a perfect square trinomial because it can be rewritten as

• \((x + 2)^2\)

This means it is derived from the square of the binomial \((x+2)\).
Identifying perfect square trinomials simplifies solving quadratic equations by allowing us to use easier algebraic methods like factoring.

Factoring

Factoring is an essential step in solving quadratic equations, particularly perfect square trinomials. Once you recognize a quadratic equation as a perfect square trinomial, you can factor it into the square of a binomial.
For example, in the equation \(x^2 + 4x + 4 = 25\), we identify it as

• \((x+2)^2 = 25\)

This simply means rewriting the quadratic in a form where it can be tackled easily through further steps, like applying other algebraic properties, which will help solve for \(x\).
Factoring is crucial because it prepares the equation for the square root property to be applied.

Square Root Property

The square root property is a useful tool for solving equations that involve squares. It states that if you have \((a)^2 = b\), then

• \(a = \pm \sqrt{b}\)

With our equation, \((x+2)^2 = 25\), we apply the square root property and find:

• \(x + 2 = \pm \sqrt{25}\)
• \(x + 2 = \pm 5\)

This step simplifies the equation and provides us potential solutions for \(x\).
Knowing how to effectively apply the square root property can greatly streamline the process of finding solutions to quadratic equations.

Simplifying Radicals

Simplifying radicals is an important process when dealing with irrational numbers in algebra. In our example, we arrive at

• \(x + 2 = \pm 5\)

Here, \(\sqrt{25}\) is already a simple, perfect square but not all equations have such easy radicals.
If the square root had been something like \(\sqrt{24}\), we would need to simplify it further into \(2\sqrt{6}\).

• This is done by factoring out perfect squares from under the radical.
• The goal is to express the radical in its simplest form.

Simplifying radicals ensures clarity in solutions and is vital for accurate computation, especially when dealing with real-world problems or more complex algebraic equations.