# Perfect Squares Made Easy: Rational Coefficient Quadratics

## Perfect Square of Quadratic Trinomials Rational Coefficients

As an experienced mathematics tutor, I’ve seen many students struggle with quadratic trinomials, especially when it comes to perfect squares with rational coefficients. However, mastering this concept is crucial for success in algebra and beyond. In this article, we’ll break down the process of identifying and solving perfect square quadratic trinomials with rational coefficients, providing you with the tools and confidence needed to tackle these problems with ease.

## Understanding Perfect Square Quadratic Trinomials

Before diving into rational coefficients, let’s review what perfect square quadratic trinomials are. A perfect square trinomial is a polynomial that can be factored into two identical binomials. In other words, it’s the square of a binomial.

The general form of a perfect square quadratic trinomial is:

$a^2 + 2ab + b^2$

where $a$ and $b$ can be any terms, including variables, constants, or a combination of both.

### Identifying Perfect Squares

To identify a perfect square quadratic trinomial, look for the following characteristics:

- The first and last terms are perfect squares.
- The middle term is twice the product of the square roots of the first and last terms.

If a quadratic trinomial meets these criteria, it can be factored as a perfect square.

## Rational Coefficients in Perfect Square Quadratic Trinomials

Now that we’ve reviewed perfect square trinomials, let’s focus on those with rational coefficients. Rational coefficients are fractions or ratios of integers, such as $\displaystyle \frac{1}{2}$, $\displaystyle \frac{3}{4}$, or $\displaystyle -\frac{5}{6}$.

When a perfect square quadratic trinomial has rational coefficients, the process of factoring remains the same, but the terms may involve fractions.

### Example 1: Identifying a Perfect Square with Rational Coefficients

Consider the following quadratic trinomial:

$\displaystyle \frac{1}{4}x^2 + x + 1$

Let’s check if it meets the criteria for a perfect square:

- The first term, $\displaystyle \frac{1}{4}x^2$, is a perfect square: $\displaystyle \frac{1}{2}x \times \frac{1}{2}x = \frac{1}{4}x^2$.
- The last term, 1, is a perfect square: $1 \times 1 = 1$.
- The middle term, $x$, is twice the product of the square roots of the first and last terms: $2 \times \displaystyle \frac{1}{2}x \times 1 = x$.

Since the trinomial meets all the criteria, it is a perfect square.

### Example 2: Factoring a Perfect Square with Rational Coefficients

Now, let’s factor the perfect square quadratic trinomial from Example 1.

$\displaystyle \frac{1}{4}x^2 + x + 1$

Step 1: Identify the square roots of the first and last terms.

- Square root of $\displaystyle \frac{1}{4}x^2$ is $\displaystyle \frac{1}{2}x$.
- Square root of $1$ is $1$.

Step 2: Write the binomial by combining the square roots and the appropriate sign ($+$ or $-$).

$\displaystyle \left(\frac{1}{2}x + 1 \right)^2$

Therefore, $\displaystyle \frac{1}{4}x^2 + x + 1 = \left(\frac{1}{2}x + 1 \right)^2$.

## Solving Perfect Square Quadratic Trinomials with Rational Coefficients

Once you’ve factored a perfect square quadratic trinomial, solving the equation becomes much easier.

### Example 3: Solving a Perfect Square Quadratic Trinomial Equation

Let’s solve the equation: $\displaystyle \frac{1}{9}x^2-\frac{2}{3}x + 1 = 0$

Step 1: Identify and factor the perfect square trinomial.

- First term: $\displaystyle \frac{1}{9}x^2 = \left(\frac{1}{3}x \right)^2$
- Last term: $1 = 1^2$
- Middle term: $\displaystyle -\frac{2}{3}x = -2 \times \frac{1}{3}x \times 1$

The trinomial is a perfect square: $\displaystyle \left(\frac{1}{3}x-1 \right)^2 = 0$

Step 2: Solve the equation by taking the square root of both sides. $\displaystyle \frac{1}{3}x-1 = 0$

Step 3: Solve the linear equations. $\displaystyle \frac{1}{3}x = 1$

Therefore, the solutions to the equation are $x = 3$.

## Practice Problems

To reinforce your understanding of perfect square quadratic trinomials with rational coefficients, try solving these practice problems:

- Factorise: $\displaystyle \frac{9}{16}x^2 + \frac{3}{2}x + 1$
- Solve: $\displaystyle \frac{1}{25}x^2 + \frac{2}{5}x + 1 = 0$
- Factorise: $\displaystyle \frac{4}{9}x^2-\frac{4}{3}x + 1$
- Solve: $\displaystyle \frac{1}{16}x^2-\frac{1}{2}x + 1 = 0$

## Conclusion

Perfect square quadratic trinomials with rational coefficients may seem intimidating at first, but with practice and understanding, you can master this concept. Remember to identify the characteristics of a perfect square, factor the trinomial, and solve the resulting equation.

By following the steps outlined in this article and practising with the given examples, you’ll be well on your way to confidently tackling perfect square quadratic trinomials with rational coefficients. This skill will serve you well in algebra and beyond, as quadratic equations appear in various mathematical contexts.

Keep practising, and don’t hesitate to seek help from your teacher or tutor if you need further guidance. With dedication and perseverance, you’ll soon find yourself mastering perfect squares with rational coefficients and excelling in your mathematics education.

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