Question

Solve the equation by completing the square. $$x^{2}+5 x-\frac{11}{4}=0$$

Short answer

\(x = -2.5 \pm \sqrt{\frac{11}{4} + 6.25}\)

Step-by-step solution

- Rewrite the equation

First, rewrite the equation to isolate the term with x, such that the equation becomes \(x^2 + 5x = \frac{11}{4}\).

- Complete the square

To complete the square, we take half of the coefficient of x, square it, and add it to both sides of the equation. Here, the coefficient of x is 5, so half of 5 is \(2.5\), and \(2.5^2\) is \(6.25\). So, the equation becomes \(x^2 + 5x + 6.25 = \frac{11}{4} + 6.25\).

- Factorize the left side

The left side of the equation is now a perfect square trinomial that can be factorized into \((x + 2.5)^2\). Hence, the equation becomes \((x + 2.5)^2 = \frac{11}{4} + 6.25\).

- Solve for x

Finally, apply the square root to both sides and solve for x by isolating it. The solutions are \(x = -2.5 \pm \sqrt{\frac{11}{4} + 6.25}\).

Key concepts

Algebraic Equations

Algebraic equations are like puzzles that mathematicians love to solve. Their pieces consist of numbers, variables (like our friend 'x'), and operations (such as addition and multiplication). The goal is to find the value of the unknown variable that makes the equation true. The equation in our exercise, , requires finding a specific value for 'x'. To unlock 'x', you'll often move pieces around using basic algebraic operations – similar to rearranging puzzle pieces until you see the whole picture. When equations get complicated, there's a neat trick called completing the square that often helps to solve them, and it's the technique we use to solve the given equation.

Factoring Trinomials

Factoring is essentially breaking down a complex expression into simpler parts or factors that, when multiplied together, give back the original expression. When we have a trinomial, which is a polynomial with three terms, factoring becomes a bit like detective work, trying to find the two numbers that work just right. In the case of the equation provided, , after completing the square, our detective work turns out to be simple since it becomes a perfect square trinomial. Think of it as if we've found a match – the factors are mirror images of each other, and they fit together perfectly to form the squared term .

Perfect Square Trinomial

So, what's a perfect square trinomial? It's a special kind of trinomial that can be expressed as the square of a binomial. Imagine squaring a binomial like , and you get a trinomial . In our exercise, completing the square transforms the left side of the equation into the format of , which we recognize as a perfect square trinomial. It's perfect because it follows a specific pattern where the first and last terms are perfect squares themselves, and the middle term is twice the product of the square roots of these perfect squares. By recognizing this pattern, we can neatly reverse engineer (or factor) the expression back into its binomial form, completing the square and paving the way to find the value of 'x'.