Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares. The zero-factor property is then used to find solutions. See
[link] ,
[link] , and
[link] .
Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See
[link] and
[link] .
Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See
[link] and
[link] .
Completing the square is a method of solving quadratic equations when the equation cannot be factored. See
[link].
A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See
[link] .
The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See
[link].
The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See
[link].
Section exercises
Verbal
How do we recognize when an equation is quadratic?
It is a second-degree equation (the highest variable exponent is 2).
When we solve a quadratic equation, how many solutions should we always start out seeking? Explain why when solving a quadratic equation in the form
$\text{\hspace{0.17em}}a{x}^{2}+bx+c=0\text{\hspace{0.17em}}$ we may graph the equation
$\text{\hspace{0.17em}}y=a{x}^{2}+bx+c\text{\hspace{0.17em}}$ and have no zeroes (
x -intercepts).
When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?
We want to take advantage of the zero property of multiplication in the fact that if
$\text{\hspace{0.17em}}a\cdot b=0\text{\hspace{0.17em}}$ then it must follow that each factor separately offers a solution to the product being zero:
$\text{\hspace{0.17em}}a=0\text{}or\text{b}=0.$
In the quadratic formula, what is the name of the expression under the radical sign
$\text{\hspace{0.17em}}{b}^{2}-4ac,$ and how does it determine the number of and nature of our solutions?
Describe two scenarios where using the square root property to solve a quadratic equation would be the most efficient method.
One, when no linear term is present (no
x term), such as
$\text{\hspace{0.17em}}{x}^{2}=16.\text{\hspace{0.17em}}$ Two, when the equation is already in the form
$\text{\hspace{0.17em}}{(ax+b)}^{2}=d.$
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.