Solving Quadratic Equations By Factoring A Step-by-Step Guide

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Factoring is a powerful technique for solving quadratic equations. It involves expressing the quadratic expression as a product of two linear factors. When a quadratic equation is factored, we can use the zero-product property to find the solutions. 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 this article, we will explore how to solve quadratic equations by factoring through several examples. We will cover the basic steps involved in factoring quadratic expressions and applying the zero-product property to find the solutions.

1. Solving $4x^2 - 12x + 5 = 0$ by Factoring

To solve the quadratic equation $4x^2 - 12x + 5 = 0$ by factoring, we need to find two binomials that multiply to give the quadratic expression. This process involves several steps, including identifying the coefficients, finding factor pairs, and testing different combinations until the correct factorization is found. Factoring quadratic equations is a fundamental skill in algebra and is crucial for solving many types of mathematical problems. By mastering this technique, students can gain a deeper understanding of algebraic concepts and improve their problem-solving abilities.

Step-by-step Factoring Process

The given quadratic equation is $4x^2 - 12x + 5 = 0$. First, we look for two numbers that multiply to the product of the leading coefficient (4) and the constant term (5), which is 4 * 5 = 20, and add up to the middle coefficient (-12). The two numbers that satisfy these conditions are -10 and -2 because (-10) * (-2) = 20 and (-10) + (-2) = -12. Now, we rewrite the middle term (-12x) using these two numbers:

4x2−10x−2x+5=04x^2 - 10x - 2x + 5 = 0

Next, we factor by grouping. We group the first two terms and the last two terms:

(4x2−10x)+(−2x+5)=0(4x^2 - 10x) + (-2x + 5) = 0

Factor out the greatest common factor (GCF) from each group. From the first group, the GCF is 2x, and from the second group, the GCF is -1:

2x(2x−5)−1(2x−5)=02x(2x - 5) - 1(2x - 5) = 0

Notice that both terms now have a common factor of (2x - 5). We factor this out:

(2x−5)(2x−1)=0(2x - 5)(2x - 1) = 0

Applying the Zero-Product Property

Now that we have factored the quadratic equation, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations:

2x - 5 = 0$ or $2x - 1 = 0

Solving the first equation, we add 5 to both sides:

2x=52x = 5

Divide by 2:

x=52x = \frac{5}{2}

Solving the second equation, we add 1 to both sides:

2x=12x = 1

Divide by 2:

x=12x = \frac{1}{2}

Therefore, the solutions to the quadratic equation $4x^2 - 12x + 5 = 0$ are $x = \frac{5}{2}$ and $x = \frac{1}{2}$. These values are the roots of the equation, and they represent the points where the parabola described by the quadratic equation intersects the x-axis. Understanding how to find these roots is crucial for analyzing the behavior of quadratic functions and solving related problems in algebra and calculus. Factoring is one of the most efficient methods for finding these roots when the quadratic expression can be easily factored.

2. Factoring $y^2 - 10y + 21 = 0$ to Find Solutions

The quadratic equation $y^2 - 10y + 21 = 0$ can also be solved by factoring. This equation is a standard quadratic form, and factoring it involves finding two numbers that multiply to the constant term (21) and add up to the coefficient of the linear term (-10). The ability to factor quadratic equations like this is a fundamental skill in algebra. It allows us to solve equations that model various real-world phenomena, such as projectile motion, optimization problems, and geometric relationships. By mastering factoring techniques, students can enhance their problem-solving capabilities and gain a deeper understanding of mathematical concepts.

Step-by-step Factoring Process

To factor the quadratic equation $y^2 - 10y + 21 = 0$, we need to find two numbers that multiply to 21 and add up to -10. The factor pairs of 21 are (1, 21), (3, 7), (-1, -21), and (-3, -7). Among these pairs, -3 and -7 satisfy both conditions because (-3) * (-7) = 21 and (-3) + (-7) = -10. Now, we can rewrite the quadratic expression as a product of two binomials:

(y−3)(y−7)=0(y - 3)(y - 7) = 0

Applying the Zero-Product Property

Now we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations:

y - 3 = 0$ or $y - 7 = 0

Solving the first equation, we add 3 to both sides:

y=3y = 3

Solving the second equation, we add 7 to both sides:

y=7y = 7

Thus, the solutions to the quadratic equation $y^2 - 10y + 21 = 0$ are $y = 3$ and $y = 7$. These solutions represent the values of y that make the equation true. In the context of a quadratic function, these values are the x-intercepts of the parabola. Understanding how to find these intercepts is crucial for graphing quadratic functions and analyzing their behavior. Factoring is an efficient method for finding these intercepts when the quadratic expression can be easily factored.

3. Solving $x^2 + 18x = 40$ by Factoring

To solve the equation $x^2 + 18x = 40$ by factoring, the first step is to rewrite the equation in the standard quadratic form, which is $ax^2 + bx + c = 0$. This involves moving all terms to one side of the equation, leaving zero on the other side. Once the equation is in standard form, we can proceed with factoring. Solving quadratic equations by factoring is a fundamental skill in algebra. It allows us to find the values of the variable that make the equation true. These solutions are also known as the roots or zeros of the quadratic equation. Mastering this technique is essential for solving a wide range of mathematical problems and applications.

Rewriting the Equation in Standard Form

To rewrite the equation $x^2 + 18x = 40$ in standard form, we subtract 40 from both sides:

x2+18x−40=0x^2 + 18x - 40 = 0

Factoring the Quadratic Expression

Now we need to factor the quadratic expression $x^2 + 18x - 40$. We look for two numbers that multiply to -40 and add up to 18. The factor pairs of -40 are:

  • (1, -40)
  • (-1, 40)
  • (2, -20)
  • (-2, 20)
  • (4, -10)
  • (-4, 10)
  • (5, -8)
  • (-5, 8)

Among these pairs, -2 and 20 satisfy the conditions because (-2) * 20 = -40 and (-2) + 20 = 18. Thus, we can factor the quadratic expression as:

(x−2)(x+20)=0(x - 2)(x + 20) = 0

Applying the Zero-Product Property

Next, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations:

x - 2 = 0$ or $x + 20 = 0

Solving the first equation, we add 2 to both sides:

x=2x = 2

Solving the second equation, we subtract 20 from both sides:

x=−20x = -20

Therefore, the solutions to the quadratic equation $x^2 + 18x = 40$ are $x = 2$ and $x = -20$. These solutions are the x-intercepts of the parabola represented by the quadratic equation. Understanding how to find these intercepts is crucial for graphing quadratic functions and analyzing their behavior. Factoring is a powerful tool for finding these solutions, especially when the quadratic expression can be easily factored. By mastering this technique, students can solve a wide range of mathematical problems and applications involving quadratic equations.

Conclusion

In conclusion, solving quadratic equations by factoring is a fundamental skill in algebra that involves expressing the quadratic expression as a product of two linear factors and applying the zero-product property. We have demonstrated this technique through several examples, including $4x^2 - 12x + 5 = 0$, $y^2 - 10y + 21 = 0$, and $x^2 + 18x = 40$. Each example illustrates the step-by-step process of factoring, from identifying the coefficients and finding factor pairs to applying the zero-product property and finding the solutions. Factoring is not only an efficient method for solving quadratic equations but also provides a deeper understanding of algebraic concepts and enhances problem-solving abilities. By mastering this technique, students can tackle a wide range of mathematical problems and applications involving quadratic equations. The solutions found through factoring represent the roots or zeros of the quadratic equation, which are crucial for analyzing the behavior of quadratic functions and their graphs. Therefore, a strong grasp of factoring is essential for success in algebra and beyond.