By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.
Completing the square is a mathematical technique used to rewrite a quadratic expression in the form of a perfect square trinomial. It involves manipulating the expression to create a squared binomial, which can be useful for solving equations, graphing functions, and simplifying expressions.
Completing the square is a fundamental concept in algebra and is used extensively in various fields, including physics, engineering, and economics. In data analysis, it is used to model and analyze quadratic relationships, such as the motion of objects under the influence of gravity or the growth of populations. For example, in physics, the equation of motion for an object under gravity can be written as $$s(t) = -\frac{1}{2}gt^2 + v_0t + s_0,$$ where $s(t)$ is the position of the object at time $t$, $g$ is the acceleration due to gravity, $v_0$ is the initial velocity, and $s_0$ is the initial position. By completing the square, we can rewrite this equation in the form $s(t) = a(t-h)^2 + k$, where $a$, $h$, and $k$ are constants.
A quadratic expression is a polynomial of degree 2, which can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
A perfect square trinomial is a quadratic expression that can be written in the form $(x + h)^2 = x^2 + 2hx + h^2$, where $h$ is a constant.
The formula for completing the square is $$x^2 + bx = (x + \frac{b}{2})^2 - \frac{b^2}{4}.$$
To complete the square, follow these steps:
Complete the square for the expression $x^2 + 6x$.
$$\begin{align} x^2 + 6x &= (x + \frac{6}{2})^2 - \frac{6^2}{4} \ &= (x + 3)^2 - 9 \end{align}$$
Complete the square for the expression $x^2 - 4x + 3$.
$$\begin{align} x^2 - 4x + 3 &= (x - \frac{4}{2})^2 - \frac{(-4)^2}{4} + 3 \ &= (x - 2)^2 - 4 + 3 \ &= (x - 2)^2 - 1 \end{align}$$
Complete the square for the expression $x^2 + 2x - 5$.
$$\begin{align} x^2 + 2x - 5 &= (x + \frac{2}{2})^2 - \frac{2^2}{4} - 5 \ &= (x + 1)^2 - 1 - 5 \ &= (x + 1)^2 - 6 \end{align}$$
What is the formula for completing the square?
A) $x^2 + bx = (x + \frac{b}{2})^2 + \frac{b^2}{4}$ B) $x^2 + bx = (x + \frac{b}{2})^2 - \frac{b^2}{4}$ C) $x^2 + bx = (x - \frac{b}{2})^2 + \frac{b^2}{4}$ D) $x^2 + bx = (x - \frac{b}{2})^2 - \frac{b^2}{4}$
B) $x^2 + bx = (x + \frac{b}{2})^2 - \frac{b^2}{4}$
The correct answer is B) $x^2 + bx = (x + \frac{b}{2})^2 - \frac{b^2}{4}$ because this is the correct formula for completing the square.
What is the value of h in the expression $x^2 + 6x$?
A) $h = \frac{6}{2}$ B) $h = \frac{6}{4}$ C) $h = \frac{4}{6}$ D) $h = \frac{2}{6}$
A) $h = \frac{6}{2}$
The correct answer is A) $h = \frac{6}{2}$ because this is the value of h in the expression $x^2 + 6x$.
What is the result of completing the square for the expression $x^2 + 2x - 5$?
A) $(x + 1)^2 - 6$ B) $(x - 1)^2 - 6$ C) $(x + 2)^2 - 6$ D) $(x - 2)^2 - 6$
A) $(x + 1)^2 - 6$
The correct answer is A) $(x + 1)^2 - 6$ because this is the result of completing the square for the expression $x^2 + 2x - 5$.
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