1. Rewrite The Equation $x^2 - 6 = 16x + 30$ In Standard Form. 2. Factor The Equation $x^2 - 6 = 16x + 30$ After Rewriting It In Standard Form. 3. What Are The Solutions For $x$ In The Equation $x^2 - 6 = 16x + 30$?

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In mathematics, solving equations is a fundamental skill. Among the various types of equations, quadratic equations hold a significant place due to their frequent appearance in various mathematical and real-world applications. One of the most common and effective methods for solving quadratic equations is factoring. This article will delve into the process of solving quadratic equations by factoring, providing a step-by-step guide and illustrative examples.

1. Rewriting the Equation in Standard Form

Before factoring a quadratic equation, it is crucial to rewrite it in its standard form. The standard form of a quadratic equation is expressed as:

ax2+bx+c=0ax^2 + bx + c = 0

where a, b, and c are constants, and x is the variable. The coefficient a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. Rearranging the equation into standard form allows us to easily identify the coefficients and constant term, which are essential for the factoring process.

Let's consider the example equation given:

x2−6=16x+30x^2 - 6 = 16x + 30

To rewrite this equation in standard form, we need to move all terms to one side, leaving zero on the other side. Here’s how we do it:

  1. Subtract 16x16x from both sides:

    x2−16x−6=30x^2 - 16x - 6 = 30

  2. Subtract 3030 from both sides:

    x2−16x−36=0x^2 - 16x - 36 = 0

Now, the equation is in standard form: x2−16x−36=0x^2 - 16x - 36 = 0. This form allows us to clearly see that a=1a = 1, b=−16b = -16, and c=−36c = -36. This rearrangement is a foundational step, as the standard form provides a clear structure for the subsequent factoring process. Without this initial step, the factoring process can become significantly more complex and prone to errors.

2. Factoring the Equation

Once the quadratic equation is in standard form, the next step is to factor it. Factoring involves expressing the quadratic expression as a product of two binomials. The general form of factoring a quadratic equation is:

ax2+bx+c=(px+q)(rx+s)ax^2 + bx + c = (px + q)(rx + s)

where p, q, r, and s are constants. In simpler cases where a=1a = 1, the factoring process becomes more straightforward. We look for two numbers that multiply to c (the constant term) and add up to b (the linear coefficient). Let's apply this to our example equation:

x2−16x−36=0x^2 - 16x - 36 = 0

Here, we need to find two numbers that multiply to −36-36 and add up to −16-16. These two numbers are −18-18 and 22, since (−18)imes2=−36(-18) imes 2 = -36 and (−18)+2=−16(-18) + 2 = -16. Therefore, we can factor the equation as:

(x−18)(x+2)=0(x - 18)(x + 2) = 0

This factorization step is crucial. It transforms the quadratic equation from a single complex expression into a product of two simpler expressions. This transformation is the key to finding the solutions, as it allows us to apply the zero-product property. Understanding how to factor quadratic expressions is a fundamental skill in algebra, and proficiency in this area can greatly simplify the process of solving quadratic equations.

Techniques for Factoring

Factoring quadratic equations involves various techniques, each suited for different types of expressions. Here are a few common methods:

  1. Greatest Common Factor (GCF): Always start by looking for the greatest common factor that can be factored out from all terms in the equation. This simplifies the equation and makes subsequent factoring easier.

  2. Difference of Squares: If the equation is in the form a2−b2a^2 - b^2, it can be factored as (a−b)(a+b)(a - b)(a + b).

  3. Perfect Square Trinomials: Equations in the form a2+2ab+b2a^2 + 2ab + b^2 or a2−2ab+b2a^2 - 2ab + b^2 can be factored as (a+b)2(a + b)^2 or (a−b)2(a - b)^2, respectively.

  4. Trial and Error: For more complex quadratic equations, trial and error involves systematically testing different combinations of factors until the correct one is found.

  5. Factoring by Grouping: This technique is used for quadratic equations with four terms. Terms are grouped in pairs, and common factors are factored out from each pair.

Understanding and applying these techniques is essential for efficient factoring. The ability to quickly identify the appropriate factoring method can save time and reduce errors in solving quadratic equations.

3. Finding the Values of x

After factoring the quadratic equation, the next step is to find the values of x that make the equation true. This is where the zero-product property comes into play. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if AB=0AB = 0, then A=0A = 0 or B=0B = 0 (or both).

Applying this property to our factored equation (x−18)(x+2)=0(x - 18)(x + 2) = 0, we set each factor equal to zero and solve for x:

  1. x−18=0x - 18 = 0

    Add 18 to both sides:

    x=18x = 18

  2. x+2=0x + 2 = 0

    Subtract 2 from both sides:

    x=−2x = -2

Thus, the solutions to the quadratic equation x2−16x−36=0x^2 - 16x - 36 = 0 are x=18x = 18 and x=−2x = -2. These values of x are the roots or zeros of the quadratic equation. They represent the points where the parabola represented by the quadratic equation intersects the x-axis. Finding these values is the ultimate goal of solving the equation, as they provide the specific values of the variable that satisfy the equation.

Verification of Solutions

It is always a good practice to verify the solutions by substituting them back into the original equation to ensure they are correct. Let's verify our solutions:

  1. For x=18x = 18:

    Original equation: x2−6=16x+30x^2 - 6 = 16x + 30

    Substitute x=18x = 18:

    (18)2−6=16(18)+30(18)^2 - 6 = 16(18) + 30

    324−6=288+30324 - 6 = 288 + 30

    318=318318 = 318 (True)

  2. For x=−2x = -2:

    Original equation: x2−6=16x+30x^2 - 6 = 16x + 30

    Substitute x=−2x = -2:

    (−2)2−6=16(−2)+30(-2)^2 - 6 = 16(-2) + 30

    4−6=−32+304 - 6 = -32 + 30

    −2=−2-2 = -2 (True)

Both solutions satisfy the original equation, confirming that our factoring and solution process was correct. Verification is a crucial step in solving equations, as it helps to catch any errors that may have occurred during the process. It provides confidence in the accuracy of the solutions and ensures that they are valid.

Conclusion

Solving quadratic equations by factoring is a fundamental skill in algebra. This method involves rewriting the equation in standard form, factoring the quadratic expression into two binomials, and then applying the zero-product property to find the values of x that satisfy the equation. Understanding and mastering this process is essential for success in more advanced mathematical concepts. By following the step-by-step guide outlined in this article and practicing with various examples, one can develop proficiency in solving quadratic equations by factoring. This skill is not only valuable in mathematics but also in various real-world applications where quadratic equations arise.

By consistently practicing and applying these steps, solving quadratic equations by factoring can become a straightforward and efficient process. This method is a cornerstone of algebraic problem-solving and provides a solid foundation for tackling more complex equations in the future.