Equations Quadratic in Form
In a quadratic equation we have a variable and its square (x and x2). An equation
that contains an expression and the square of that expression is quadratic in form
if substituting a single variable for that expression results in a quadratic equation.
Equations that are quadratic in form can be solved by using methods for quadratic
equations.
Example
An equation quadratic in form
Solve (x + 15)2 - 3(x + 15) - 18 = 0
Solution
Note that x + 15 and (x + 15)2 both appear in the equation. Let a
= x + 15 and
substitute a for x + 15 in the equation:
| (x + 15)2
- 3(x + 15) - 18 |
= 0 |
|
| a2 - 3a - 18 |
= 0 |
|
| (a - 6)(a + 3) |
= 0 |
Factor. |
| a - 6 |
= 0 |
or |
a + 3 |
= 0 |
|
| a |
= 6 |
or |
a |
= -3 |
|
| x + 15 |
= 6 |
or |
x + 15 |
= -3 |
Replace a by x + 15. |
| x |
= -9 |
or |
x |
= -18 |
|
Check in the original equation. The solution set is {-18, -9}.
|
