Solve The Quadratic Equation Using Factoring

Solve The Quadratic Equation Using Factoring

Analyze and solve quadratic equations by employing factoring techniques, including making appropriate substitutions where necessary. Explain your calculation process thoroughly and demonstrate understanding of quadratic solutions.

Paper For Above instruction

Solving quadratic equations is a fundamental skill in algebra that reflects the ability to manipulate and interpret polynomial expressions. Among various solution strategies, factoring remains one of the most straightforward and insightful methods for quadratic equations, especially when the quadratic can be expressed as a product of binomials. This paper discusses techniques including direct factoring, substitution methods, and their applications, supported by detailed solution processes and explanations.

Introduction

Quadratic equations are polynomial equations of degree two, generally expressed in the form ax² + bx + c = 0, where a, b, and c are coefficients, with a ≠ 0. The goal is to find the values of x that satisfy the equation, known as roots or solutions. Factoring is a practical method to solve quadratics when the trinomial factors neatly into binomials. When factoring directly is not immediately feasible, substitution methods can be employed, especially when the quadratic involves quadratic expressions in terms of x², such as in equations like x⁴ - 16x² = -60.

Factorization of Quadratic Equations

To illustrate, consider the quadratic equation 15x² - x = 2. First, set the equation to zero:

15x² - x - 2 = 0.

Next, look for two numbers that multiply to 15 (-2) = -30 and add to -1. These numbers are -6 and 5 because (-6) 5 = -30, and (-6) + 5 = -1.

Rewrite the middle term using these numbers:

15x² - 6x + 5x - 2 = 0.

Group and factor out common terms:

(15x² - 6x) + (5x - 2) = 0,

which simplifies to:

3x(5x - 2) + 1(5x - 2) = 0.

Factor out the common binomial factor:

(3x + 1)(5x - 2) = 0.

Set each factor equal to zero and solve:

3x + 1 = 0 → x = -1/3,

5x - 2 = 0 → x = 2/5.

Therefore, the solutions are x = -1/3 and x = 2/5.

Using Substitution: Quadratic in Terms of x²

Another common scenario involves quadratic equations in the form x⁴ - 16x² = -60. To solve such equations, substitution simplifies the process. Let u = x²; then, the equation becomes:

u² - 16u = -60.

Bring all to one side:

u² - 16u + 60 = 0.

This quadratic in u can be factored or solved via the quadratic formula. Factoring factors of 60 that sum to -16 are -10 and -6:

(u - 10)(u - 6) = 0.

Set each factor equal to zero:

u - 10 = 0 → u = 10,

u - 6 = 0 → u = 6.

Recall that u = x²; then, substitute back:

x² = 10 → x = ±√10,

x² = 6 → x = ±√6.

The solutions in x are thus x = ±√10 and x = ±√6. These provide four real solutions, considering positive and negative roots.

Graphical Interpretation and Domain & Range

Understanding the domain and range of functions provides crucial insights into the behavior of quadratic functions. For example, the function y = f(x) can be analyzed using its graph. If the graph is a parabola opening upward or downward, the domain typically covers all real numbers, as the x-values extend infinitely in both directions. The range depends on the vertex of the parabola and the direction it opens.

For instance, y = –x² + 7x + 50 reaches a maximum at its vertex. The domain of this quadratic is all real numbers, as quadratic functions are defined everywhere. The range is from the minimum value observed at the vertex to infinity or negative infinity depending on the parabola’s orientation.

Transformations and Algebra of Functions

Transformations of functions are shifts, stretches, compressions, and reflections that alter the graph’s position and shape without changing its fundamental nature. For functions like f(x) = –3x – 3 and g(x) = 4x² + 2x - 2, algebraic operations such as sum, difference, and composition can be performed.

For example, adding a constant shifts the graph vertically; multiplying by a scalar stretches or compresses the graph vertically. Factoring quadratic functions assists in understanding roots and symmetry, facilitating graph sketching and solving related equations.

Conclusion

Factoring is an essential and effective method for solving quadratic equations, providing clear insight into the roots of polynomials. When direct factoring is not apparent, substitution techniques, especially for higher-degree quadratics or equations involving x², are invaluable. Understanding the graphical representations of these functions enriches comprehension and enables better interpretation of solutions. Mastery of these methods enhances problem-solving skills in algebra and broader mathematical contexts.

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