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In a quadratic equation we have a variable and its square (x and x^{2}). 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

a^{2} - 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}.