Solving (x^2 + 6x + 90 = 0) with the Quadratic Formula
Quadratic equations are the backbone of algebra and appear in countless real‑world problems—from projectile motion to finance. When you encounter an equation of the form
[ ax^2 + bx + c = 0, ]
the quadratic formula gives you the exact values of (x) that satisfy the equation. In this article we’ll walk through the process of applying the formula to the specific quadratic
[ x^2 + 6x + 90 = 0, ]
explore the meaning of its discriminant, discuss why the solutions are complex, and show how the formula fits into the broader context of solving quadratics.
1. Recap: The Quadratic Formula
For any quadratic equation (ax^2 + bx + c = 0) (with (a \neq 0)), the solutions are
[ x = \frac{-b \pm \sqrt{,b^2 - 4ac,}}{2a}. ]
The expression under the square root, (b^2 - 4ac), is called the discriminant. It tells us:
- If the discriminant > 0: two distinct real roots.
- If the discriminant = 0: one real root (a repeated root).
- If the discriminant < 0: two complex conjugate roots.
2. Identifying (a), (b), and (c)
For the equation (x^2 + 6x + 90 = 0):
| Coefficient | Symbol | Value |
|---|---|---|
| Quadratic term | (a) | 1 |
| Linear term | (b) | 6 |
| Constant term | (c) | 90 |
3. Calculating the Discriminant
[ \Delta = b^2 - 4ac = 6^2 - 4(1)(90) = 36 - 360 = -324. ]
Because (\Delta) is negative, the equation has no real solutions; instead, it has two complex conjugate solutions.
4. Applying the Quadratic Formula
Insert the values into the formula:
[ x = \frac{-6 \pm \sqrt{-324}}{2(1)} = \frac{-6 \pm \sqrt{-324}}{2}. ]
Since (\sqrt{-324} = i\sqrt{324} = i \cdot 18 = 18i) (where (i) is the imaginary unit), we get
[ x = \frac{-6 \pm 18i}{2} = -3 \pm 9i. ]
Thus, the two complex solutions are:
- (x_1 = -3 + 9i)
- (x_2 = -3 - 9i)
These are conjugate pairs, as expected when the discriminant is negative.
5. Interpreting the Results
- Graphical View: The parabola (y = x^2 + 6x + 90) opens upwards (since (a = 1 > 0)). Its vertex is at ((-3, 81)), far above the (x)-axis. So, the curve never crosses the axis, explaining why there are no real roots.
- Real‑World Context: In physics, a negative discriminant could indicate that a projectile with given initial conditions will never reach a particular height, or in economics that a quadratic cost function never yields a zero profit under the modeled parameters.
- Complex Numbers: The appearance of (i) shows that the equation can be solved within the complex plane, providing a complete solution set even when real solutions are absent.
6. Alternative Methods (Quick Check)
While the quadratic formula is universal, other techniques can also solve this equation:
-
Completing the Square
[ x^2 + 6x + 90 = (x^2 + 6x + 9) + 81 = (x+3)^2 + 81 = 0. ] Then ((x+3)^2 = -81) → (x+3 = \pm 9i) → (x = -3 \pm 9i) It's one of those things that adds up.. -
Factoring (Not Possible Here)
Since the discriminant is negative, the polynomial cannot factor over the reals.
These methods confirm the same complex roots as the quadratic formula Worth keeping that in mind..
7. Common Mistakes to Avoid
| Mistake | Why It Happens | Fix |
|---|---|---|
| Using the wrong sign in the numerator | Confusing (-b + \sqrt{\Delta}) with (-b - \sqrt{\Delta}) | Remember the (\pm) symbol: compute both possibilities. |
| Ignoring the negative discriminant | Assuming a real root exists | Recognize that (\sqrt{-k} = i\sqrt{k}) and work in the complex field. |
| Misplacing parentheses | Misinterpreting the order of operations | Always write the formula as (\frac{-b \pm \sqrt{\Delta}}{2a}). |
| Forgetting to divide by (2a) | Forgetting the denominator after simplifying the numerator | Double‑check that the entire numerator is divided by (2a). |
8. Extending the Concept: Complex Conjugates
When a quadratic has complex roots, they always come in conjugate pairs: (p + qi) and (p - qi). This property ensures that the coefficients of the original polynomial remain real. In our example, the roots (-3 \pm 9i) satisfy this rule.
9. Practical Applications
- Engineering: Determining natural frequencies in vibration analysis often leads to quadratics with complex roots, indicating oscillatory behavior.
- Computer Graphics: Calculating intersection points of curves may involve solving quadratics; complex solutions mean the curves do not intersect in real space.
- Control Systems: Characteristic equations of linear systems are quadratic; complex roots correspond to damped oscillations.
10. FAQ
| Question | Answer |
|---|---|
| Can the quadratic formula handle equations where (a = 0)? | The formula yields a single real root: (x = -b/(2a)). Which means ** |
| **What if the discriminant is zero? ** | Graphing will show the parabola never crossing the (x)-axis, confirming no real roots. |
| Can I solve (x^2 + 6x + 90 = 0) by graphing? | No. If (a = 0), the equation is linear, not quadratic. |
| **Why do we use (i) instead of (\sqrt{-1})? | |
| Do complex roots always occur in pairs? | (i) is the standard notation for the imaginary unit, simplifying calculations. |
11. Conclusion
The quadratic formula is a powerful, universal tool that guarantees a solution for any quadratic equation, whether the roots are real or complex. On the flip side, understanding the discriminant not only tells us about the nature of the roots but also guides us toward alternative solving methods such as completing the square. This leads to by applying it to (x^2 + 6x + 90 = 0), we discovered that the equation has no real solutions but two complex conjugate roots, (-3 \pm 9i). Mastery of these concepts equips you to tackle a wide array of algebraic problems, whether in pure mathematics, physics, engineering, or beyond Not complicated — just consistent..
The quadratic formula’s enduringrelevance lies in its ability to adapt to the complexities of mathematical problems, whether they involve real or complex solutions. Its structured approach—rooted in
the discriminant—removes the guesswork from solving second-degree polynomials and provides a reliable roadmap for students and professionals alike. By bridging the gap between basic algebra and complex number theory, it transforms a "no solution" scenario into a deeper exploration of the imaginary plane.
The bottom line: the transition from real to complex roots is more than just a mathematical trick; it is a fundamental shift in perspective that allows us to describe phenomena that are otherwise invisible on a standard Cartesian graph. Whether you are analyzing the trajectory of a projectile or the stability of an electrical circuit, the quadratic formula serves as the essential bridge between the theoretical equation and the physical reality it represents. By mastering its application and understanding the nuances of the discriminant, you gain a versatile skill set that is indispensable for any advanced study in the STEM fields.
12. Extending the Idea: Quadratics in Higher Dimensions
Although the classic quadratic equation lives in one‑dimensional algebra, the same principles appear in multivariate contexts. Consider a quadratic form in two variables,
[ Q(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f, ]
where (a,b,c,d,e,f\in\mathbb{R}). Setting (Q(x,y)=0) yields a conic section (ellipse, parabola, hyperbola, or degenerate case). The discriminant now involves the determinant of the associated matrix:
[ \Delta = \begin{vmatrix} a & \tfrac{b}{2}\[4pt] \tfrac{b}{2} & c \end{vmatrix}=ac-\frac{b^{2}}{4}. ]
Just as the scalar discriminant (,b^{2}-4ac) tells us whether a one‑dimensional quadratic has real or complex roots, this matrix discriminant tells us about the shape and orientation of the conic. For instance:
| (\Delta) sign | Conic type |
|---|---|
| (\Delta>0) & Ellipse (or circle if (a=c) and (b=0)) | |
| (\Delta=0) & Parabola (degenerate when also (a=c=0)) | |
| (\Delta<0) & Hyperbola |
Thus, the notion of “real vs. complex” roots generalises to “real vs. imaginary intersections” of a curve with the plane. In practice, solving (Q(x,y)=0) often reduces to completing the square in both variables or applying a rotation that diagonalises the quadratic form—techniques that echo the single‑variable completing‑the‑square method we used earlier.
13. Computational Tips for the Quadratic Formula
When you implement the quadratic formula in a programming language or a spreadsheet, numerical stability becomes a concern, especially when (b^{2}) is much larger than (4ac). A classic trick is to compute the roots as follows:
import math
def quadratic_roots(a, b, c):
disc = b*b - 4*a*c
sqrt_disc = math.Because of that, sqrt(abs(disc))
if disc >= 0:
# real roots
q = -0. 5 * (b + math.
The key idea is to avoid subtracting two nearly equal large numbers, which can cause catastrophic cancellation. By computing \(q = -\frac{1}{2}\bigl(b + \operatorname{sgn}(b)\sqrt{b^{2}-4ac}\bigr)\) first, we preserve precision for the smaller root.
### 14. Historical Aside: From Babylon to Cardano
Quadratic equations predate the formula we use today. Clay tablets from Babylon (circa 1800 BC) show solutions to equations of the form \(x^{2}+px = q\) by geometric methods. The Greeks, notably Euclid, used constructions with straightedge and compass to find roots of specific quadratics. That said, it was not until the Islamic Golden Age that al‑Khwarizmi (c. 780–850) gave a systematic algorithm for solving \(ax^{2}+bx=c\) by completing the square—an approach that directly leads to the modern formula.
The leap to a closed‑form expression occurred in the 16th century. Italian mathematicians such as Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano published solutions for the general quadratic while simultaneously wrestling with the “impossible” square roots that would later be recognised as complex numbers. Their work laid the groundwork for the acceptance of \(i\) as a legitimate mathematical entity, a development that finally allowed the quadratic formula to be stated without caveat.
### 15. Real‑World Applications
| Field | Typical Quadratic Problem | Why the Formula Matters |
|------|---------------------------|--------------------------|
| **Physics** | Projectile motion: \(y = -\frac{g}{2v_{x}^{2}}x^{2}+ \tan\theta\,x + y_{0}\) | Determines range, maximum height, and impact point. |
| **Finance** | Present value of an annuity leading to \(ax^{2}+bx+c=0\) | Solves for interest rates or payment periods. |
| **Engineering** | Damped harmonic oscillator: \(m\ddot{x}+c\dot{x}+kx=0\) → characteristic equation \(mx^{2}+cx+k=0\) | Predicts under‑, over‑, or critically‑damped behavior. In real terms, |
| **Computer Graphics** | Ray‑sphere intersection: \(\| \mathbf{o}+t\mathbf{d} - \mathbf{c} \|^{2}=r^{2}\) → quadratic in \(t\) | Determines visibility and shading in ray tracing. |
| **Biology** | Logistic growth with harvesting: \(rN\left(1-\frac{N}{K}\right)-H=0\) reduces to a quadratic in \(N\) | Finds sustainable population levels.
In each case, the discriminant instantly tells the practitioner whether a feasible solution exists (e.g., a projectile will hit the ground, a ray will intersect an object) and, if so, how many.
### 16. A Final Word on Intuition
When you first encounter the quadratic formula, it can feel like a mechanical recipe: plug numbers, compute, and obtain answers. Yet, each component carries geometric and physical meaning:
- **\(b\)** shifts the parabola left or right; \(-b/(2a)\) locates its vertex.
- **\(c\)** raises or lowers the entire curve.
- **\(a\)** controls the “tightness” of the opening.
- **\(\sqrt{b^{2}-4ac}\)** measures the vertical distance from the vertex to the x‑axis; when this distance is imaginary, the parabola never meets the axis.
By visualising these effects, the formula transforms from a rote calculation into a diagnostic tool that reveals the shape and behavior of quadratic relationships at a glance.
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## Conclusion
The quadratic formula stands as one of mathematics’ most elegant bridges between algebraic abstraction and concrete problem‑solving. By dissecting the equation \(x^{2}+6x+90=0\) we observed how the discriminant dictates the transition from real to complex roots, how completing the square offers an alternative pathway, and how these ideas echo in multivariate quadratic forms, computational practice, and real‑world models across science and engineering.
Not the most exciting part, but easily the most useful.
Understanding not just *how* to apply the formula, but *why* each term matters, equips you with a versatile lens for interpreting any second‑degree relationship you encounter. Whether you are charting the flight of a basketball, designing a stable control system, or rendering a 3‑D scene, the quadratic formula will continue to provide a reliable, insight‑rich method for uncovering the hidden roots that shape our world.