What is the result of substituting a quadratic expression into another equation?

When substituting a quadratic expression into another equation, the result is a new quadratic equation. This is because a quadratic expression is generally written in the form ( ax^2 + bx + c ), where ( a ), ( b ), and ( c ) are constants and ( a \neq 0 ).

When this quadratic expression is placed into another equation—in particular, if that equation is also polynomial—it results in terms that maintain the highest power of ( x^2 ), thus preserving the quadratic nature. In some cases, if you were substituting into a linear equation, you would still find a quadratic equation since the substitution can lead to ( x ) being isolated in a quadratic form.

For example, if the quadratic expression ( (x + 2)^2 ) is substituted into a linear equation such as ( y = x ), the new equation becomes ( y = (x + 2)^2 ), which expands to ( y = x^2 + 4x + 4 ), a quadratic equation. Therefore, substituting a quadratic expression into another equation indeed generates a new quadratic equation, reflecting the original quadratic's characteristics.