- Rearrange the equation into the standard form \(9x^2 + 30x + 25 = 0\).
- Recognize the expression as a perfect square trinomial \((3x + 5)^2 = 0\).
- Solve the equation \(3x + 5 = 0\) to find the root.
- The final answer is \(x = \boxed{-\frac{5}{3}}\).
Explanation
- Problem Analysis
The problem asks us to solve the quadratic equation \(-30x - 25 = 9x^2\) for \(x\). Our goal is to rearrange this into the standard form \(ax^2 + bx + c = 0\) and then determine the roots.
- Standard Form and Factoring
To put the equation in standard form, we move all terms to one side:
\[9x^2 + 30x + 25 = 0\]
Here, the coefficients are \(a = 9\), \(b = 30\), and \(c = 25\). We can now look to factor this quadratic expression. Notice that it follows the perfect square trinomial pattern \(A^2 + 2AB + B^2 = (A + B)^2\). Identifying \(A = 3x\) and \(B = 5\), we have:
\[(3x)^2 + 2(3x)(5) + 5^2 = 0\]
\[(3x + 5)^2 = 0\]
- Solving for x
Since \((3x + 5)^2 = 0\), the only solution is found by setting the base to zero:
\[3x + 5 = 0\]
Subtracting 5 from both sides gives:
\[3x = -5\]
Dividing by 3 yields:
\[x = -\frac{5}{3}\]
Thus, the equation has exactly one solution.
Examples
This problem involves solving a quadratic equation, a fundamental skill in algebra used in various real-life scenarios. For example, if you are designing a parabolic arch for a bridge, the equation for the height of the arch relative to its base can be modeled as a quadratic function. Solving for the roots tells you exactly where the arch meets the ground. Understanding how to factor these expressions allows engineers to quickly determine key dimensions in construction and physics.