Problem 37 - 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 \(x = -6 + 3\sqrt{7}\) and \(x = -6 - 3\sqrt{7}\).

Step 1: Identify the Quadratic Equation

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a=1\), \(b=12\), and \(c=-27\).

Step 2: Calculate the Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute in the values: \[D = 12^2 - 4 \times 1 \times (-27) = 144 + 108 = 252\].

Step 3: Determine the Nature of the Roots

Since the discriminant \(D = 252 > 0\), the quadratic equation has two distinct real roots.

Step 4: Calculate the Roots Using the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) to find the roots, where \(a=1\), \(b=12\), and \(D=252\). First, calculate \(\sqrt{252}\), which simplifies to \(6\sqrt{7}\). Thus, the roots are: \[x_1 = \frac{-12 + 6\sqrt{7}}{2} = -6 + 3\sqrt{7}\] and \[x_2 = \frac{-12 - 6\sqrt{7}}{2} = -6 - 3\sqrt{7}\].

Key Concepts

Discriminant

When solving quadratic equations, understanding the discriminant is crucial. The discriminant, represented by the letter \( D \), is part of the quadratic formula that helps us determine the nature of the roots of a quadratic equation. A quadratic equation is generally given by \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The discriminant formula is \( D = b^2 - 4ac \).

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one real root (also known as a repeated root).
• If \( D < 0 \), the equation has no real roots, but two complex roots.

For the given quadratic equation \( x^2 + 12x - 27 = 0 \), substituting the values \( a = 1 \), \( b = 12 \), and \( c = -27 \) into the discriminant formula gives \( D = 12^2 - 4 \times 1 \times (-27) = 252 \). This positive discriminant indicates that the equation has two distinct real roots.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). If you ever come across a quadratic equation that cannot be easily factored, the quadratic formula is your go-to method. The formula is:\[x = \frac{-b \pm \sqrt{D}}{2a}\]Where:

• \(x\) represents the roots (solutions) of the quadratic equation.
• \(b\) is the coefficient of \(x\).
• \(D\) is the discriminant, \( b^2 - 4ac \).

Using the given quadratic formula with \( a = 1 \), \( b = 12 \), and the previously calculated \( D = 252 \), you apply the quadratic formula as follows:

• Calculate \( \sqrt{252} \), which simplifies to \( 6\sqrt{7} \).
• Substitute into the formula to find the roots:
  • \( x_1 = \frac{-12 + 6\sqrt{7}}{2} = -6 + 3\sqrt{7} \)
  • \( x_2 = \frac{-12 - 6\sqrt{7}}{2} = -6 - 3\sqrt{7} \)

Real Solutions

Real solutions are the values of \( x \) that satisfy the quadratic equation and are also real numbers, as opposed to imaginary numbers. In other words, real solutions are those that we can plot on the number line and make sense in the real world. The discriminant helps us quickly determine whether a quadratic equation has real solutions, thanks to its value:

• If \( D > 0 \), there are two distinct real solutions because the square root of a positive number is a real number.
• If \( D = 0 \), there's exactly one real solution, due to the single repeated root causing the term \( \pm 0 \) in the quadratic formula.

For the equation \( x^2 + 12x - 27 = 0 \), with \( D = 252 \), we already determined it has two distinct real roots because the discriminant is greater than zero. Thus, the roots \( -6 + 3\sqrt{7} \) and \( -6 - 3\sqrt{7} \) are both real numbers and serve as the complete set of real solutions to the equation.