Question
What is the quadratic equation that was asked to be solved using the perfect square trinomial pattern?
Answers
GPT-4o mini
To solve a quadratic equation using the perfect square trinomial pattern, we first need to recognize the form of such an equation. A quadratic expression can be expressed as a perfect square trinomial in the form:
\[
(ax + b)^2 = c
\]
This expands to:
\[
a^2x^2 + 2abx + b^2 = c
\]
To solve a quadratic equation using this pattern, you would typically rearrange the equation such that one side equals zero and attempt to express the left-hand side as a perfect square.
For instance, consider the quadratic equation:
\[
x^2 + 6x + 9 = 0
\]
This can be recognized as a perfect square trinomial:
\[
(x + 3)^2 = 0
\]
To solve for \(x\), we can take the square root of both sides:
\[
x + 3 = 0
\]
Thus, the solution is:
\[
x = -3
\]
Therefore, a quadratic equation that can be solved using the perfect square trinomial is \(x^2 + 6x + 9 = 0\), which factors to \((x + 3)^2 = 0\).
\[
(ax + b)^2 = c
\]
This expands to:
\[
a^2x^2 + 2abx + b^2 = c
\]
To solve a quadratic equation using this pattern, you would typically rearrange the equation such that one side equals zero and attempt to express the left-hand side as a perfect square.
For instance, consider the quadratic equation:
\[
x^2 + 6x + 9 = 0
\]
This can be recognized as a perfect square trinomial:
\[
(x + 3)^2 = 0
\]
To solve for \(x\), we can take the square root of both sides:
\[
x + 3 = 0
\]
Thus, the solution is:
\[
x = -3
\]
Therefore, a quadratic equation that can be solved using the perfect square trinomial is \(x^2 + 6x + 9 = 0\), which factors to \((x + 3)^2 = 0\).