91Ó°ÊÓ

A quadratic equation is a type of polynomial equation that involves a squared term and can be written in the general form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). This characteristic square term \( ax^2 \) gives the quadratic equation its parabolic shape when graphed, forming a U-shaped curve either upwards or downwards depending on the sign of \( a \).
Quadratic equations often arise in various real-world scenarios such as calculating area, projectile motion, and consumer economics. Because of their wide application, learning to solve quadratic equations is an important skill. The equation can be solved by various methods, including:

• Factoring
• Using the quadratic formula
• Completing the square

Simplification through algebraic substitution can be especially useful when dealing with complex terms.

Factoring is one of the methods used to solve quadratic equations. It involves expressing the equation as a product of its factors. Once a quadratic equation is factorable, it can be set to zero and solved by determining the values that make each factor zero.
To factor a quadratic equation like \( y^2 - 11y + 24 = 0 \), you need to find two numbers that multiply to 24 (the constant term) and add up to -11 (the coefficient of \( y \)). In this case, the numbers -3 and -8 fit the criteria, leading to the factors \((y - 3)(y - 8) = 0\).
This means that the roots of the equation are \( y = 3 \) and \( y = 8 \). Factoring is efficient when the quadratic is neatly factorable, making quick work of finding solutions if the correct pair is identified.

When factoring is not straightforward or feasible, the quadratic formula is a reliable tool for finding the roots of a quadratic equation. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). The expression under the square root, \( b^2 - 4ac \), is called the discriminant, and it determines the nature of the roots:

• If the discriminant is positive, there are two real and distinct roots.
• If it is zero, there are two real and identical roots.
• If it is negative, the roots are complex or imaginary.

The quadratic formula can always be applied, making it a universal method for solving quadratics even when other methods fail.

Equation solving often involves a step-by-step strategy of simplifying and manipulating expressions to find unknown values. Quadratic equations, in particular, require careful analysis since they can have complex solutions.
The solved exercise showcases these fundamental steps:

• First, substitution was used: \( y = x^2 - 2x \) simplifies the original equation to \( y^2 - 11y + 24 \).
• Next, solving \( y^2 - 11y + 24 = 0 \) through factoring gives potential solutions for \( y \).
• Then substitute back to determine \( x \) from the obtained values of \( y \), applying the quadratic formula to solve the resulting equations.

This process ensures that all possible solutions are explored methodically. By practicing different techniques, you’ll become adept at selecting the best approach for each unique equation.