3.07 Further quadratic equations
Quadratic functions are used widely in science, business and engineering to model physical phenomena and how quantities change over time. The physical parabolic shape of a quadratic can describe the movement of water in a fountain, the movement of a ball (and in fact any object thrown into the air), or the shape of satellite dishes.
When solving practical problems these steps might be helpful:
- Identify the unknown (what are you solving for?) and use clues given to express other relevant quantities in terms of the unknown. Sometimes you will have to introduce a pronumeral to represent the unknown amount.
- Use the practical context to find relationships between the different amounts involved, and construct equations.
- Solve them, often for when they equal zero, or to find the maximum or minimum (more on these types of problems in later chapters).
- Use your common sense and the context of the problem to interpret the results. For example, when finding length, area or time, can these have negative values?
When at the solution stage of the problem useful starting points are:
- Does the equation have to be rearranged to get to a more workable form?
- Can a common factor be removed immediately?
- Can a variable substitution be made?
Then we employ one of our solving methods:
- Solve using algebraic manipulation - For quadratics such as $2x^2-1=49$2x2−1=49.
- Factorise - Fully factorising a quadratic means we can then use the null factor law. If $a\times b=0$a×b=0 then either $a=0$a=0 or $b=0$b=0.
- Completing the square - This method gets us to a point where we can then solve algebraically, for example $\left(x-3\right)^2=16$(x−3)2=16.
- Quadratic Formula - This method will solve any quadratic equation of the form $ax^2+bx+c=0$ax2+bx+c=0, but it is not always the most time effective if a method above can be utilised.
- Technology - Once we have extracted the important information from a question in context and formed an equation, we could use technology to solve the equation. Don't forget to interpret the result in the context of the problem.
Practice questions
Question 1
Solve the following equation:
$x-\frac{45}{x}=4$x−45x=4
Write all solutions on the same line, separated by commas.
Question 2
The area of a rectangle is $160$160 square metres. If the length of the rectangle is $6$6 m longer than its width, find the dimensions.
Let $w$w be the width of the rectangle in metres.
First find the width of the rectangle by solving for $w$w.
Now find the rectangle's length.
Question 3
The product of two consecutive, positive odd numbers is $399$399. Let the smaller number be $2x+1$2x+1.
First solve for $x$x.
Now find the first odd number.
Lastly, find the other odd number.
Question 4
Solve the following equation for $x$x by substituting in $m=4^x$m=4x.
$4^{2x}-65\times4^x+64=0$42x−65×4x+64=0
Use a comma to separate multiple solutions.