91Ó°ÊÓ

A quadratic equation is a type of polynomial equation of the form \( ax^{2} + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). This equation represents a parabola when graphed on a coordinate plane. Quadratic equations are essential in various fields, including physics, engineering, and economics because they model scenarios involving projectile motions, area calculations, and more.
The quadratic equation in our exercise is \( x^{2} - 3x = 18 \). To bring it into standard form, simply move all terms to one side, resulting in \( x^{2} - 3x - 18 = 0 \). Here, \( a = 1 \), \( b = -3 \), and \( c = -18 \). Recognizing that you have a quadratic equation is the first step in solving it.

The quadratic formula provides a reliable method for identifying the roots of a quadratic equation. This formula is given by: \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\] where \( a, b, \) and \( c \) are the coefficients from our quadratic equation. Use this formula when you cannot factorize the equation easily or when precision is essential.
In our example, we substituted \( a = 1 \), \( b = -3 \), and \( c = -18 \) into the quadratic formula, giving us: \[x = \frac{-(-3) \pm \sqrt{(-3)^{2} - 4 \times 1 \times (-18)}}{2 \times 1}\] Simplifying this, we calculate the discriminant \( b^{2} - 4ac \) which is equal to 81. This number under the square root sign determines the nature and number of solutions. A positive discriminant means two real solutions exist.

Mathematical problem-solving involves using logical reasoning and established procedures to find solutions to complex problems. In the case of quadratic equations, the quadratic formula is a systematic approach to determine the roots.
Once the discriminant is calculated, convert it to a real number— in this situation, the square root of 81 equals 9. Substitute this back into the formula resulting in two possible solutions:

• \( x = \frac{3 + 9}{2} = 6 \)
• \( x = \frac{3 - 9}{2} = -3 \)

By following these mathematical steps, we ensure that our solutions are correct and precise. This process is crucial for successfully tackling a variety of mathematical problems, reinforcing your future understanding and capability to handle more advanced equations.