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Magazine Chapter 8: Problem 62
Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ g(x)=x^{2}+5 x+2 $$
Short Answer
Expert verified
The x-intercepts are \( x = \frac{-5 + \sqrt{17}}{2} \) and \( x = \frac{-5 - \sqrt{17}}{2} \).
Step by step solution
01
Write in standard form
Start with the given equation in standard quadratic form: \( g(x) = x^2 + 5x + 2 \).
02
Move constant term to the other side
Next, we want to isolate the quadratic and linear terms on one side. Subtract 2 from both sides: \( g(x) - 2 = x^2 + 5x \). For finding x-intercepts, set \(g(x)\) to 0: \(0 - 2 = x^2 + 5x \) which simplifies to \(x^2 + 5x = -2\).
03
Prepare to complete the square
To complete the square, take the coefficient of \( x \) (which is 5), divide it by 2, and then square it. \ \( \left(\frac{5}{2}\right)^2 = \frac{25}{4} \). Add and subtract this square inside the equation: \( x^2 + 5x + \frac{25}{4} - \frac{25}{4} = -2 \).
04
Form the perfect square trinomial
Rewrite the left-hand side as a perfect square trinomial: \( \left(x + \frac{5}{2}\right)^2 - \frac{25}{4} = -2 \).
05
Simplify the equation
Add \( \frac{25}{4} \) to both sides to isolate the perfect square term: \( \left(x + \frac{5}{2}\right)^2 = -2 + \frac{25}{4} \). Combine the terms on the right-hand side: \( \left(x + \frac{5}{2}\right)^2 = \frac{25}{4} - \frac{8}{4} = \frac{17}{4} \).
06
Solve for x
Take the square root of both sides: \( x + \frac{5}{2} = \pm \sqrt{\frac{17}{4}} \). Simplify: \( x + \frac{5}{2} = \pm \frac{\sqrt{17}}{2} \). Solve for \( x \) by subtracting \( \frac{5}{2} \) from both sides: \( x = -\frac{5}{2} \pm \frac{\sqrt{17}}{2} \).
07
Write the final x-intercept solutions
Simplify the expression to find the x-intercepts: \( x = \frac{-5 + \sqrt{17}}{2} \) and \( x = \frac{-5 - \sqrt{17}}{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
quadratic functions
Quadratic functions are a type of polynomial function where the highest degree of any term is 2. This means their general form is shown as \[f(x) = ax^2 + bx + c\].
Here, \(a\), \(b\), and \(c\) are constants, with \(x\) representing the variable. The graph of a quadratic function is a curve called a parabola.
It can open upwards if \(a > 0\) or downwards if \(a < 0\). The point where the parabola changes direction is known as the vertex. Understanding the properties of quadratic functions
is essential in solving quadratic equations and finding their x-intercepts.
Here, \(a\), \(b\), and \(c\) are constants, with \(x\) representing the variable. The graph of a quadratic function is a curve called a parabola.
It can open upwards if \(a > 0\) or downwards if \(a < 0\). The point where the parabola changes direction is known as the vertex. Understanding the properties of quadratic functions
is essential in solving quadratic equations and finding their x-intercepts.
x-intercepts
The x-intercepts of a function are the points where the graph crosses the x-axis. For a quadratic function,
these are the solutions to the equation \(ax^2 + bx + c = 0\). To find these points, we can use various methods like factoring, using the quadratic formula, or completing the square.
When working with the equation \(g(x) = x^2 + 5x + 2\), setting it to zero gives the equation \(x^2 + 5x + 2 = 0\). The solutions to this equation will provide the x-intercepts of the function.
these are the solutions to the equation \(ax^2 + bx + c = 0\). To find these points, we can use various methods like factoring, using the quadratic formula, or completing the square.
When working with the equation \(g(x) = x^2 + 5x + 2\), setting it to zero gives the equation \(x^2 + 5x + 2 = 0\). The solutions to this equation will provide the x-intercepts of the function.
perfect square trinomial
A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. In other words,
\(a^2 + 2ab + b^2\) can be factored as \((a + b)^2\). This concept is critical when completing the square. For example,
to transform \(x^2 + 5x + 2\) into a perfect square trinomial, we add and subtract \(\left(\frac{5}{2}\right)^2\) to get \(x^2 + 5x + \frac{25}{4} - \frac{25}{4} = -2\).
Then we rewrite \(x^2 + 5x + \frac{25}{4}\) as \(\left(x + \frac{5}{2}\right)^2\), making it a perfect square trinomial.
\(a^2 + 2ab + b^2\) can be factored as \((a + b)^2\). This concept is critical when completing the square. For example,
to transform \(x^2 + 5x + 2\) into a perfect square trinomial, we add and subtract \(\left(\frac{5}{2}\right)^2\) to get \(x^2 + 5x + \frac{25}{4} - \frac{25}{4} = -2\).
Then we rewrite \(x^2 + 5x + \frac{25}{4}\) as \(\left(x + \frac{5}{2}\right)^2\), making it a perfect square trinomial.
square roots
Finding square roots is an important skill in algebra and especially in solving quadratic equations. When we have an equation of the form \((x + d)^2 = k\),
we solve for \(x\) by taking the square root of both sides. It results in two solutions: \(x + d = \sqrt{k}\) and \(x + d = -\sqrt{k}\).
For instance, with our equation \(\left(x + \frac{5}{2}\right)^2 = \frac{17}{4}\), taking the square root yields \(x + \frac{5}{2} = \pm \frac{\sqrt{17}}{2}\).
We then isolate \(x\) to find the specific values \(x = \frac{-5 + \sqrt{17}}{2}\) and \(x = \frac{-5 - \sqrt{17}}{2}\) as the x-intercepts of the function.
we solve for \(x\) by taking the square root of both sides. It results in two solutions: \(x + d = \sqrt{k}\) and \(x + d = -\sqrt{k}\).
For instance, with our equation \(\left(x + \frac{5}{2}\right)^2 = \frac{17}{4}\), taking the square root yields \(x + \frac{5}{2} = \pm \frac{\sqrt{17}}{2}\).
We then isolate \(x\) to find the specific values \(x = \frac{-5 + \sqrt{17}}{2}\) and \(x = \frac{-5 - \sqrt{17}}{2}\) as the x-intercepts of the function.
