- Rewrite the equation as \(x^2 + 12x - 9 = 0\).
- Identify coefficients \(a = 1\), \(b = 12\), \(c = -9\) and apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- Simplify the discriminant to find \(x = \frac{-12 \pm 6\sqrt{5}}{2}\).
- Obtain the solution \(x = -6 \pm 3\sqrt{5}\), which corresponds to the second option.
- Final answer: \(\boxed{x = -6 \pm 3\sqrt{5}}\).
Explanation
- Analysis of the quadratic equation
We are tasked with solving the quadratic equation \(x^2 + 12x = 9\). To solve this, we first bring the equation into the standard quadratic form \(ax^2 + bx + c = 0\) by subtracting 9 from both sides:
\[x^2 + 12x - 9 = 0\]
From this standard form, we can identify the coefficients: \(a = 1\), \(b = 12\), and \(c = -9\).
- Applying the quadratic formula
We can solve for \(x\) using the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Substituting the identified coefficients \(a = 1\), \(b = 12\), and \(c = -9\) into the formula, we get:
\[x = \frac{-12 \pm \sqrt{12^2 - 4(1)(-9)}}{2(1)}\]
\[x = \frac{-12 \pm \sqrt{144 + 36}}{2}\]
\[x = \frac{-12 \pm \sqrt{180}}{2}\]
- Simplifying the result and identifying the correct option
Now we simplify the discriminant \(\sqrt{180}\). Note that \(180 = 36 \times 5\), so \(\sqrt{180} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}\).
Substituting this back into our expression for \(x\):
\[x = \frac{-12 \pm 6\sqrt{5}}{2}\]
Dividing both terms in the numerator by 2, we obtain:
\[x = -6 \pm 3\sqrt{5}\]
This result matches the second option provided in the problem.
Examples
Quadratic equations are fundamental in physics and engineering. For example, if you throw a ball, its height \(h\) at time \(t\) can be modeled by a quadratic equation \(h(t) = -gt^2 + v_0t + h_0\). By setting \(h(t) = 0\), you can solve for the time \(t\) when the ball hits the ground using the quadratic formula, just as we did here to find the roots of the given equation. Understanding how to solve these equations allows us to predict the trajectory of projectiles, analyze economic growth patterns, and design structures like bridges and arches.