Problem 50 - Introductory Algebra for College Students

Question

Determine whether statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In completing the square for \(x^{2}-7 x=5,\) we should add \(\frac{49}{4}\) to both sides.

Step-by-Step Solution

Answer

The statement provided in the exercise is True. No changes are required.

Step 1: Verification

To verify the statement, first let's identify the coefficient of x in the given equation, which is -7. The process of 'completing the square' involves adding \(\frac{B^{2}}{4}\) to both sides of the equation, where B is the coefficient of x. Thus, for the given equation, it will be \((-7/2)^{2}\) which equals \(\frac{49}{4}\).

Step 2: Performing Operation

Now, add \(\frac{49}{4}\) to both sides of the given equation, resulting in: \(x^{2}-7 x + \frac{49}{4} = 5 + \frac{49}{4}.\)

Step 3: Conclusion

On observing, the left side of the equation is now a perfect square. Thus, the statement used in the exercise is found to be correct. There's no need for any changes to the statement.

Key Concepts

Algebra and Completing the Square

Algebra is a broad division of mathematics that deals with symbols and the rules for manipulating those symbols. It's like a shorthand language to express mathematical ideas, including the properties of numbers and the relationships between them. When we talk about solving quadratic equations in algebra, one potent technique is 'completing the square.' This method allows us to transform a quadratic equation, which typically looks like ax^2 + bx + c = 0, into a form involving a perfect square trinomial. This transformation makes the equation easier to solve.

Completing the square is particularly useful because not all quadratics can be factored easily. Taking the equation x^2 - 7x = 5, as in our original exercise, we can complete the square by finding a number to add to both sides that allows us to represent the equation as (x - h)^2 = k, where h and k are constants. In this process, 'h' is half of the coefficient of 'x', squared, added to both sides. This method not only simplifies quadratic equations but also lays the foundation for understanding the vertex form of a parabola and even the derivation of the quadratic formula.

Quadratic Equations and Their Solutions

Quadratic equations can be recognized by their general form a x^2 + b x + c = 0, where 'a', 'b', and 'c' are constants, and 'a' isn't zero. The goal when solving these equations is to find the value(s) of 'x' that make the equation true. There are several ways to solve quadratic equations, like factoring, using the quadratic formula, graphing, or completing the square, which we're focusing on.

Completing the square transforms the original quadratic equation into a perfect square trinomial, making it straightforward to solve by taking the square root of both sides. Subsequently, you isolate 'x' to find its possible values. This method is impeccable for situations where factoring becomes complicated or isn't feasible. It’s vital because it often provides the clearest path to the solutions, which is the primary objective in algebra – to find the unknowns.

Perfect Square Trinomials

Understanding perfect square trinomials is fundamental when completing the square in algebra. A perfect square trinomial is a special kind of polynomial, always the result of squaring a binomial. In general, a binomial squared, such as (x + d)^2, results in a trinomial x^2 + 2dx + d^2, where the first and last terms are perfect squares, and the middle term is twice the product of the square roots of these perfect squares.

In the context of our exercise, adding \(\frac{49}{4}\) to both sides creates the perfect square trinomial on the left side, \(x^2 - 7x + \frac{49}{4}\), which corresponds to \(\left(x - \frac{7}{2}\right)^2\). This trinomial is now easy to solve because we simply take the square root of both sides and solve for 'x'. Recognizing and creating perfect squares are crucial for simplifying complex algebraic equations and are handy in calculus and other advanced fields of mathematics as well.