Question

Find all the real solutions of the equation. \(x^3-10 x^2+19 x+30=0\)

Short answer

The real solution of the equation is \(x = 10\).

Step-by-step solution

Identify the coefficients

The coefficients for the given cubic equation \(x^3 - 10x^2 + 19x + 30 = 0\) are: \(a = 1, b = -10, c = 19, d = 30\).

Apply Cardano's formula

Cardano's formula provides a method for finding the roots of any cubic equation. Firstly, compute \[ \Delta_0 = b^2 - 3ac = (-10)^2 - 3(1)(19) = 100 - 57 = 43,\]and \[\Delta_1 = 2b^3 - 9abc + 27a^2d = 2(-10)^3 - 9(1)(-10)(19) + 27(1)^2(30) = -2000 + 1710 + 810 = 520.\]Next, the discriminant \(\Delta = \Delta_1^2 - 4\Delta_0^3 = 520^2 - 4(43^3) = -70312\). Since the discriminant is negative, the equation has one real root and two complex roots. For a real root,\[ x = \frac{-b + \sqrt[3]{\frac{\Delta_1+\sqrt{\Delta}}{2}} + \sqrt[3]{\frac{\Delta_1-\sqrt{\Delta}}{2}}}{3a}= \frac{10 + \sqrt[3]{\frac{520+i\sqrt{70312}}{2}} + \sqrt[3]{\frac{520-i\sqrt{70312}}{2}}}{3}= 10.\]

Apply synthetic division

After finding first real root, synthetic division can be used to find the remaining roots. Performing synthetic division with 10 gives us new equation, \(x^2 - x - 3 = 0\). Applying quadratic formula for it results in the roots \(x = \frac{1 + \sqrt{13}}{2}\) and \(x = \frac{1 - \sqrt{13}}{2}\). But these roots aren't real, so the only real solution is \(x=10\).

Key concepts

Cardano's Formula

Understanding Cardano's formula is crucial when tackling cubic equations like the given equation: x^3 - 10x^2 + 19x + 30 = 0. The formula is designed to find the roots of cubic equations, which may include real and complex numbers. The formula first involves calculating certain values such as Δ0 and Δ1, which are used along with the coefficients of the cubic equation.

Once these values are computed, we find the discriminant Δ which dictates the nature of the roots. Intriguingly, if Δ is negative, it indicates the presence of one real root and a pair of complex conjugate roots. This makes Cardano's formula a powerful tool because it offers insights into the roots before even calculating them. However, one should be prepared for complex arithmetic, which can with involve imaginary numbers, represented by i, where i = \(sqrt{-1}\).

The real root from the formula can then be further processed, as we have seen in the solution, where synthetic division comes into play to find any remaining real roots.

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly handy when dividing by a linear factor, and you're seeking to find the other roots of a polynomial equation. After applying Cardano's formula to our cubic equation, we established one real root, x = 10.

With this knowledge, synthetic division is used to reduce the cubic equation to a quadratic one by 'synthetically' dividing the cubic polynomial by x - 10 (the binomial representing the real root). The remaining quadratic equation then allows us to pursue additional real roots, if any exist. Although in our problem, synthetic division resulted in a quadratic that eventually unveiled complex roots, demonstrating that x = 10 is the sole real solution for the original cubic.

Quadratic Formula

Once a cubic equation is simplified to a quadratic one, the quadratic formula becomes an essential tool to find the further roots. The formula x = \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) provides the solution to the quadratic equation ax^2 + bx + c = 0.

Our problem required us to apply the quadratic formula after synthetic division left us with a quadratic equation. The determinative part of the formula is under the square root, called the discriminant, which we'll discuss next; it determines the nature of the roots. Notably, real solutions are found when the discriminant is non-negative.

Discriminant

The discriminant is a critical element when deciphering the type of roots a quadratic equation will yield. Represented by b^2 - 4ac in the quadratic formula, it tells us whether the roots are real or complex without actually solving the equation.

A positive discriminant corresponds to two distinct real roots, while a zero value points to exactly one real root (a repeated root). A negative discriminant suggests the roots are complex and come in conjugate pairs. In our example, the discriminant of the cubic equation's derivative helped confirm one real root and two complex roots initially. Not to be confused, the quadratic equation derived after synthetic division also has its own discriminant which indicates whether its roots are real or complex.

Real and Complex Roots

In polynomial equations, roots can be either real or complex. A real root is a solution that can be plotted on a real number line, while a complex root has both a real part and an imaginary part and can't be represented on the real number line.

Cardano's formula and the discriminant aid in predicting the nature of roots for cubic equations. Although complex roots may seem daunting due to their imaginary components, they follow systematic rules and always come in pairs, known as conjugates. In our problem, the derived quadratic produced two complex solutions, concluding that the original cubic equation had only one real root, x = 10, and those two complex counterparts.