Solving Quadratic Equations: A Step-by-Step Guide For 3p^2 + 5p - 12 = 0

This article provides a comprehensive guide on how to solve the quadratic equation 3p² + 5p - 12 = 0. We will explore different methods, including factoring, using the quadratic formula, and completing the square, to find the solution set. Understanding how to solve quadratic equations is a fundamental skill in algebra and has wide applications in various fields of mathematics, science, and engineering. This step-by-step explanation will help you grasp the underlying concepts and techniques involved in finding the roots of a quadratic equation.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation are called roots or zeros, which are the values of the variable (in this case, 'p') that make the equation true. Quadratic equations can have two distinct real roots, one real root (a repeated root), or two complex roots.

In our given equation, 3p² + 5p - 12 = 0, we can identify the coefficients as follows:

• a = 3
• b = 5
• c = -12

The coefficients play a crucial role in determining the nature and values of the roots. Let's delve into different methods to solve this quadratic equation and find its solution set.

Method 1: Factoring

Factoring is a method of expressing the quadratic expression as a product of two linear factors. If we can factor the quadratic expression, we can easily find the roots by setting each factor equal to zero. The goal is to find two numbers that multiply to ac (3 * -12 = -36) and add up to b (5). These numbers will help us split the middle term (5p) and factor the expression. For this quadratic equation, the numbers are 9 and -4 because 9 * -4 = -36 and 9 + (-4) = 5.

Let's rewrite the middle term using these numbers:

3p² + 9p - 4p - 12 = 0

Now, we can factor by grouping:

p(3p + 9) - 4(p + 3) = 0

Factor out the common factor (3p+9) from the first group:

3p(p + 3) - 4(p + 3) = 0

Oops, there was an error in our previous factoring step. Let's restart from the step where we rewrote the middle term:

3p² + 9p - 4p - 12 = 0

Now we factor by grouping correctly:

3p(p + 3) - 4(p + 3) = 0

Now, we can factor out the common binomial factor (p + 3):

(3p - 4)(p + 3) = 0

Setting each factor equal to zero gives us the roots:

3p - 4 = 0 or p + 3 = 0

Solving for p:

3p = 4 => p = 4/3

p = -3

Thus, the roots of the quadratic equation are p = 4/3 and p = -3. Factoring relies on identifying the correct numbers to split the middle term, which can sometimes be challenging. However, when successful, it offers a straightforward way to solve quadratic equations.

Method 2: Using the Quadratic Formula

The quadratic formula is a general formula that provides the solutions to any quadratic equation in the form ax² + bx + c = 0. The formula is given by:

p = (-b ± √(b² - 4ac)) / (2a)

This formula is derived from the method of completing the square and can be applied to any quadratic equation, regardless of whether it can be factored easily. It is particularly useful when dealing with equations that have irrational or complex roots. The quadratic formula ensures a consistent approach to solving quadratic equations, making it a reliable tool in algebra.

For our equation, 3p² + 5p - 12 = 0, we have a = 3, b = 5, and c = -12. Plugging these values into the quadratic formula:

p = (-5 ± √(5² - 4 * 3 * -12)) / (2 * 3)

p = (-5 ± √(25 + 144)) / 6

p = (-5 ± √169) / 6

p = (-5 ± 13) / 6

Now, we calculate the two possible values for p:

p₁ = (-5 + 13) / 6 = 8 / 6 = 4/3

p₂ = (-5 - 13) / 6 = -18 / 6 = -3

As we found using factoring, the solutions are p = 4/3 and p = -3. The quadratic formula provides a systematic way to find these solutions by directly substituting the coefficients of the equation into the formula. This method is especially beneficial when factoring is difficult or impossible.

Method 3: Completing the Square

Completing the square is another method to solve quadratic equations. It involves transforming the equation into a perfect square trinomial, which can then be easily solved. This method provides a deeper understanding of the structure of quadratic equations and their solutions. Although it can be more complex than factoring or using the quadratic formula, completing the square is a valuable technique for advanced algebraic manipulations.

To complete the square, we first divide the equation by the coefficient of p² (if it's not 1). In our case, the equation is already in the form where the coefficient of p² is 3, so we will factor out 3 from the terms involving p:

3(p² + (5/3)p) - 12 = 0

Next, we want to add and subtract a value inside the parenthesis that will make the expression a perfect square. This value is (b/2a)², where b is the coefficient of p and a is the coefficient of p², which in the parenthesis is 1. So, we need to add and subtract ((5/3) / 2)² = (5/6)² = 25/36.

3(p² + (5/3)p + 25/36 - 25/36) - 12 = 0

3((p + 5/6)² - 25/36) - 12 = 0

Now, distribute the 3:

3(p + 5/6)² - 25/12 - 12 = 0

Move the constants to the other side of the equation:

3(p + 5/6)² = 25/12 + 12

3(p + 5/6)² = 25/12 + 144/12

3(p + 5/6)² = 169/12

Divide by 3:

(p + 5/6)² = 169/36

Take the square root of both sides:

p + 5/6 = ±√(169/36)

p + 5/6 = ±13/6

Solve for p:

p = -5/6 ± 13/6

So, the solutions are:

p₁ = (-5/6 + 13/6) = 8/6 = 4/3

p₂ = (-5/6 - 13/6) = -18/6 = -3

Again, we find the solutions p = 4/3 and p = -3. This method, while more intricate, provides a comprehensive understanding of the process of solving quadratic equations and can be particularly useful in certain advanced algebraic scenarios.

Solution Set

After applying three different methods—factoring, using the quadratic formula, and completing the square—we have consistently found the solutions to the quadratic equation 3p² + 5p - 12 = 0 to be p = 4/3 and p = -3. Therefore, the solution set is:

{-3, 4/3}

This set represents all the values of p that satisfy the given equation. Each method provided a unique approach to arriving at the same solutions, demonstrating the versatility of algebraic techniques in solving quadratic equations.

Conclusion

In conclusion, we have successfully solved the quadratic equation 3p² + 5p - 12 = 0 using three different methods: factoring, the quadratic formula, and completing the square. Each method offers a unique perspective and approach to finding the solutions. The solution set for the equation is {-3, 4/3}. Understanding these methods is crucial for mastering algebra and for solving more complex mathematical problems. Whether you prefer the directness of the quadratic formula, the strategic thinking required for factoring, or the comprehensive approach of completing the square, knowing these techniques will empower you to tackle a wide range of quadratic equations effectively.