Skills Review for Second-Order Linear Equations

Learning Outcomes

  • Given a function equation, find function values (outputs) for specified variables (inputs)
  • Apply the Quadratic Formula
  • Write function equations using given conditions

In the Second-Order Linear Equations section, we will look at differential equations that contain second derivatives and a dependent variable that is not raised to any powers itself. Here we will review how to plug variable inputs into equations, use the Quadratic Formula, and write function equations using given conditions.

Evaluate Functions at Variable Inputs

You likely have plenty of experience evaluating functions at constants. Functions can also be evaluated for inputs that are variables or expressions. The process is the same, but the simplified answer will contain a variable. The following example shows you how to evaluate a function for a variable input.

Example: Evaluating Functions at Variable Inputs

Evaluate [latex]f\left(x\right)={x}^{2}+3x - 4[/latex] at

  1. [latex]2[/latex]
  2. [latex]a[/latex]
  3. [latex]a+h[/latex]
  4. [latex]\frac{f\left(a+h\right)-f\left(a\right)}{h}[/latex]

In the following video, we show more examples of evaluating functions for both constant and variable inputs.

Apply the Quadratic Formula

The quadratic formula is a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.

A General Note: The Quadratic Formula

Written in standard form, [latex]a{x}^{2}+bx+c=0[/latex], any quadratic equation can be solved using the quadratic formula:

[latex]x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}[/latex]

where a, b, and c are real numbers and [latex]a\ne 0[/latex].

How To: Given a quadratic equation, solve it using the quadratic formula

  1. Make sure the equation is in standard form: [latex]a{x}^{2}+bx+c=0[/latex].
  2. Make note of the values of the coefficients and constant term, [latex]a,b[/latex], and [latex]c[/latex].
  3. Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.
  4. Calculate and solve.

Example : Solve A Quadratic Equation Using the Quadratic Formula

Solve the quadratic equation: [latex]{x}^{2}+5x+1=0[/latex].

Example: Solving a Quadratic Equation with the Quadratic Formula

Use the quadratic formula to solve [latex]{x}^{2}+x+2=0[/latex].

Try It

Solve the quadratic equation using the quadratic formula: [latex]9{x}^{2}+3x - 2=0[/latex].

Try It

Write Function Equations Using Given Conditions

(also in Module 3, Skills Review for Motion in Space)
Sometimes, to find a missing value in a function equation, you will be given an input of the function and the corresponding output. You will then plug this input and output into the function equation and find the missing value.

Example: Writing a Function Equation from given conditions

Given [latex]f(2)=-1[/latex], find the unknown value c in the function equation [latex]f(x)=3x^3-4x^2-x+c[/latex].

 

Try It

Given [latex]f(1)=5[/latex], find the unknown value c in the function equation [latex]f(x)=-2x^2+3x+c[/latex].