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Chapter 1: Problem 67

Find the value(s) of \(x\) for which \(f(x)=g(x)\). \(f(x)=x^{2}, \quad g(x)=x+2\)

Short Answer

Expert verified
The values of \(x\) for which \(f(x)=g(x)\) are \(x = 2\) and \(x = -1\)

Step by step solution

01

Set the Equations Equal to Each Other

Start by setting the functions equal to each other: \(f(x) = g(x)\) which becomes \(x^{2}=x+2\)
02

Turn the Equation into Quadratic Form

Re-arrange the equation to standard form of a quadratic equation (i.e., \(ax^{2} + bx + c = 0\)). In this case, bring \(x + 2\) to the left side, resulting in \(x^{2} -x - 2 = 0\)
03

Solve Quadratic Equation

This equation can be factored into \((x-2)(x+1)= 0\)
04

Solve for the Roots

Set each part of the factored equation to zero and solve for x, which are \(x = 2\) and \(x = -1\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equations
You might already know quadratic equations from school exercises. These equations appear often and are quite essential in algebra. They are polynomial equations of degree 2, meaning the highest power of the variable is 2. The general form is usually displayed as: \[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Quadratic equations have a characteristic curved graph known as a parabola. The primary task while dealing with these equations is to find their roots, which are the values of \(x\) when the equation equals zero. Sometimes, we need to rearrange equations to get them into this standard form before solving them.
Functions
Functions are foundational in mathematics and represent a rule that connects two variables, usually \(x\) and \(y\), or input and output. A function takes an input value, applies the rule, and delivers an output. In the case of the exercise, we have two functions:
  • \(f(x) = x^2\)
  • \(g(x) = x + 2\)
Both these functions define different relationships. The task was to find the point(s) where these relationships meet, meaning the \(x\) values where \(f(x)\) equals \(g(x)\). Understanding functions helps you grasp how variables correlate with each other under specific rules.
Factoring
Factoring is a method used to solve quadratic equations efficiently. When we factor, we essentially break down an equation into two simpler expressions whose product gives us the original equation. Consider the equation \[ x^2 - x - 2 = 0 \] This can be factored into two binomials as \[ (x-2)(x+1) = 0 \]. By doing so, we make the equation easier to solve because setting each binomial to zero gives simpler equations to handle. Factoring is often quicker and more straightforward than other solving methods and is especially convenient when dealing with whole numbers.
Roots
Roots are the solutions to the quadratic equation. They represent the values of \(x\) that satisfy the equation. In our example, after factoring \[ (x-2)(x+1) = 0 \], we need to find which \(x\) values make either \[ (x-2) = 0 \] or \[ (x+1) = 0 \] true. Solving these gives the solutions \( x = 2 \) and \( x = -1 \). These values, the roots, represent the points at which the parabola of the quadratic equation crosses the x-axis. Finding roots helps us unravel where both the functions intersect, helping to answer if and where the relationships between the equations meet.

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Most popular questions from this chapter

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\frac{3 x+4}{5} $$

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\sqrt{2 x+3} $$

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