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In this section, students learn how to find the roots of quadratic equations using factorization, exploring various examples and the mathematical reasoning behind identifying solutions that satisfy the quadratic equation.
In Chapter 4.3, students explore the solution of quadratic equations through factorization. A quadratic equation generally takes the form ax² + bx + c = 0, with a non-zero coefficient a. To find roots, students learn that any real number α is a root if substituting it into the equation satisfies it (i.e., it results in zero). The process of factorization is emphasized as a method to break down the quadratic polynomial into simpler linear factors. The section includes illustrative examples, such as finding the roots of the equations 2x² - 5x + 3 = 0 and 6x² - x - 2 = 0, demonstrating the splitting of the middle term and the significance of equating factors to zero to find the solutions. Further emphasis is placed on representing real-world problems as quadratic equations, reinforcing the practical applications of the concepts learned.
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Roots of Quadratic Equations: Values that satisfy the equation.
Factorization Method: Splitting the middle term to find roots.
Real-world Applications: Quadratic problems represented mathematically.
Discriminant: A tool to determine the type of roots.
Quadratics are neat, roots we must find, factor them fast, and use your mind!
Once upon a time, there were two friends called Roots and Factor who went on an adventure to solve equations, unlocking secrets with every factor they found.
To remember the quadratic formula: 'A-B-C, Over A's two, gives us the roots we pursue!'
Finding the roots of 2x² - 5x + 3 = 0 by factorization.
Solving 6x² - x - 2 = 0 using the factorization method.
Term: Quadratic Equation
Definition: An equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a is not zero.
An equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a is not zero.
Term: Roots
Definition: The values of x that satisfy a quadratic equation, making it equal to zero.
The values of x that satisfy a quadratic equation, making it equal to zero.
Term: Factorization
Definition: The process of breaking down an expression into simpler products, specifically into linear factors in the case of quadratic equations.
The process of breaking down an expression into simpler products, specifically into linear factors in the case of quadratic equations.
Term: Zeroes of a Polynomial
Definition: The x-values that make the polynomial equal to zero.
The x-values that make the polynomial equal to zero.
Term: Discriminant
Definition: The expression b² - 4ac that determines the nature of the roots of a quadratic equation.
The expression b² - 4ac that determines the nature of the roots of a quadratic equation.
We have \[ 5x^{2} + 3x + 2 = 0 \] Using the quadratic formula, \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where, - a = 5 - b = 3 - c = 2 Substituting the values, \[ x = \frac{-3 \pm \sqrt{3^{2} - 4 \cdot 5 \cdot 2}}{2 \cdot 5} \] This simplifies to \[ x = \frac{-3 \pm \sqrt{9 - 40}}{10} \] Indicating that the roots of the equation 5x^{2} + 3x + 2 = 0 will reveal complex solutions. We can verify the nature of the roots through the discriminant.