Problem 33 - Algebra and Trigonometry

Question

Find all real solutions of the equation. \(x^{2}+12 x-27=0\)

Step-by-Step Solution

Answer

The real solutions are \(x = -6 + \sqrt{63}\) and \(x = -6 - \sqrt{63}\).

Step 1: Understand the Equation

We need to find the real solutions of the quadratic equation \(x^2 + 12x - 27 = 0\). This is a standard quadratic equation of the form \(ax^2 + bx + c = 0\) with \(a = 1\), \(b = 12\), and \(c = -27\).

Step 2: Use the Quadratic Formula

To solve the quadratic equation \(ax^2 + bx + c = 0\), we use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, we substitute \(a = 1\), \(b = 12\), and \(c = -27\) into the formula.

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\). For our equation, it is \(12^2 - 4 \times 1 \times (-27)\). This simplifies to \(144 + 108 = 252\).

Step 4: Evaluate the Square Root of the Discriminant

The next step is to find \(\sqrt{252}\). This simplifies to \(\sqrt{4 \times 63} = 2\sqrt{63}\).

Step 5: Substitute into the Quadratic Formula

Plug the values (\(b = 12\), \(\pm \sqrt{252} = \pm 2\sqrt{63}\), and \(a = 1\)) into the quadratic formula: \(x = \frac{-12 \pm 2\sqrt{63}}{2}\).

Step 6: Simplify the Expression

We simplify \(x = \frac{-12 \pm 2\sqrt{63}}{2}\) to \(x = -6 \pm \sqrt{63}\). Splitting this into two separate solutions, the real solutions are \(x = -6 + \sqrt{63}\) and \(x = -6 - \sqrt{63}\).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool in algebra that helps find the solutions of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula itself is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula offers a straightforward method to calculate the roots of any quadratic equation, providing two possible values for \(x\):

• The symbol \(\pm\) indicates that there are two solutions to consider.
• Solving the equation encompasses plugging in the values of coefficients \(a\), \(b\), and \(c\) into this formula.

Using the quadratic formula ensures that we can systematically find solutions, whether they are real or imaginary (complex). When solving quadratics, one must be careful with arithmetic and properly handling the signs in the formula, as simple mistakes can lead to incorrect results.
In our specific problem, where \(a = 1\), \(b = 12\), and \(c = -27\), using the formula efficiently helps us understand the solution path.

Discriminant

The discriminant in the quadratic formula is an essential part of identifying the nature of the equation's roots. It is represented by:

• \(b^2 - 4ac\)

This expression helps determine whether the solutions to a quadratic equation will be real or complex:

• If the discriminant is positive, as in our problem where we find it to be \(252\), the quadratic equation has two distinct real solutions.
• If it equals zero, the equation has exactly one real solution, indicating that both roots are the same.
• When the discriminant is negative, the quadratic has two complex solutions, not real.

In our equation, calculating the discriminant \(144 + 108 = 252\) foretells that we will have two real solutions. It’s crucial to carefully manage arithmetic steps while calculating the discriminant, as these initial values dictate further calculations in the quadratic formula.

Real Solutions

Real solutions are those which can take any numerical value on a number line and do not involve imaginary numbers. For a quadratic equation, the discriminant (\(b^2 - 4ac\)) plays a significant role in concluding if solutions are real. When the discriminant is greater than zero, the quadratic equation guarantees two distinct real solutions.
In the exercise, upon finding that our discriminant is \(252\), we affirm the solutions are real and different. Implementing the quadratic formula \(x = \frac{-12 \pm 2\sqrt{63}}{2}\), we describe the real solutions explicitly as:

• \(x = -6 + \sqrt{63}\)
• \(x = -6 - \sqrt{63}\)

Recognizing these solutions involves straightforward arithmetic, and verifying them will reassure that the real numbers we obtain satisfy the original equation \(x^2 + 12x - 27 = 0\). It's also beneficial to double-check work here, as simple missteps can lead to incorrect conclusions.

Square Root

Taking the square root is a necessary step when using the quadratic formula. The square root operation can transform complex expressions into simpler radicals:

• In our formula, it appears as \(\sqrt{b^2 - 4ac}\).

Square roots can sometimes be simplified further through prime factorization. For instance, in our quadratic equation, after calculating the discriminant as \(252\), we simplify \(\sqrt{252}\) by recognizing it as \(\sqrt{4 \times 63} = 2\sqrt{63}\). Breaking it down step-by-step prevents calculation errors:

• Identify any perfect squares; here, 4 is a perfect square.
• Rewrite and simplify it to the form of \(2\sqrt{63}\), making it easier to use in subsequent formulas.

In simplifying square roots, careful factorization and elimination allow easier calculations and provide accuracy when applying these values in the quadratic formula. It’s crucial to ensure complete simplification for clarity in final solutions.