Problem Solving with Quadratics
Example 1
When $16$ is subtracted from the square of a number, the result is six times the original number.
Let the number be $x$.
The question can be written: $=$
The number is or
Example 2
The base of a triangle is $7\;cm$ longer than its height. If the area of the triangle is $15\;cm^2$, find the length of the base.
Let the height be $x$.
The base can be written:
The question can be written: $=15$
The base is $cm$.
Example 3
In a right-angled triangle, the hypotenuse is $1$ more than $2$ times the shortest side. The other side is $1$ less than $2$ times the shortest side. Find the length of the hypotenuse.
Let the length of the short side be $x$.
The length of the hypotenuse can be written: .
The length of the other side can be written: .
Using Pythagoras' theorem, $=\left(2x+1\right)^2$.
The length of the hypotenuse is .
Example 4
You want to make a rectangular dog park. One side is an existing wall and the other 3 sides will be made using a fence. The fence is a total of $22\;m$ and the area of the dog park is $48\;m^2$. Find the width.
Let the width be $x$.
The amount of fence left for the remaining two sides is .
The length can be written: .
The area of the dog park can be written: $=48$.
The width is $cm$ or $cm$.
