Complete The Equivalent Equation For $-7x - 60 = X^2 + 10x$. $(x + \square)(x + \square) = 0$. What Are The Solutions Of $-7x - 60 = X^2 + 10x$? $x = \square$

Jun 19, 2025 by ADMIN 159 views

In mathematics, solving equations is a fundamental skill. This article will guide you through the process of finding the solutions to a given equation, focusing on quadratic equations. We will specifically address the equation $-7x - 60 = x^2 + 10x$ , showing you how to rewrite it in the standard quadratic form, factor it, and then determine the solutions. This step-by-step approach will not only help you solve this particular problem but also equip you with the tools to tackle similar equations in the future. Understanding these methods is crucial for success in algebra and beyond, as quadratic equations appear in various real-world applications, from physics to engineering.

Rewriting the Equation

Our initial equation is $-7x - 60 = x^2 + 10x$ . To solve this, the first crucial step is to rewrite it into the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$ . This form allows us to easily apply various methods for finding solutions, such as factoring, completing the square, or using the quadratic formula. To achieve this standard form, we need to move all terms to one side of the equation, leaving zero on the other side. This involves performing algebraic operations that maintain the equality, such as adding or subtracting the same terms from both sides. By rearranging the terms, we set the stage for identifying the coefficients $a$ , $b$ , and $c$ , which are essential for further steps in solving the equation.

To rewrite the equation $-7x - 60 = x^2 + 10x$ in the standard form, we need to move all terms to one side. Let's add $7x$ and $60$ to both sides of the equation. This gives us:

$-7x - 60 + 7x + 60 = x^2 + 10x + 7x + 60$

Simplifying both sides, we get:

$0 = x^2 + 17x + 60$

Thus, the equation in standard form is:

$x^2 + 17x + 60 = 0$

This is a quadratic equation in the standard form $ax^2 + bx + c = 0$ , where $a = 1$ , $b = 17$ , and $c = 60$ . The next step is to factor this quadratic equation, which will help us find the solutions for $x$ . Factoring involves finding two binomials that, when multiplied together, give us the quadratic expression. This process relies on identifying two numbers that add up to the coefficient of the $x$ term (in this case, 17) and multiply to the constant term (in this case, 60). This transformation is a cornerstone technique in algebra, enabling us to simplify complex expressions and reveal the underlying structure of the equation. The correct factorization not only provides the solutions directly but also enhances our understanding of the equation's behavior and properties.

Factoring the Quadratic Equation

Now that we have the quadratic equation in the standard form $x^2 + 17x + 60 = 0$ , we can proceed with factoring. Factoring is the process of expressing the quadratic expression as a product of two binomials. This method is particularly effective when the quadratic expression can be easily factored, making it a straightforward way to find the solutions. The key to factoring lies in identifying two numbers that satisfy specific conditions related to the coefficients of the quadratic equation. These numbers must add up to the coefficient of the $x$ term (in our case, 17) and multiply to the constant term (in our case, 60). Finding these numbers is often a process of trial and error, but with practice, it becomes more intuitive. Once we identify these numbers, we can easily write the factored form of the quadratic equation, which directly leads to the solutions.

To factor the quadratic equation $x^2 + 17x + 60 = 0$ , we need to find two numbers that multiply to 60 and add up to 17. Let's list the factor pairs of 60:

1 and 60
2 and 30
3 and 20
4 and 15
5 and 12
6 and 10

Among these pairs, 5 and 12 add up to 17. Therefore, we can rewrite the quadratic equation as:

$(x + 5)(x + 12) = 0$

This factored form is equivalent to the original quadratic equation. The advantage of having the equation in this form is that it allows us to easily find the solutions for $x$ . The solutions are the values of $x$ that make each factor equal to zero, as the product of the factors will be zero only if at least one of them is zero. The ability to factor quadratic equations is a vital skill in algebra, opening the door to solving a wide range of problems. It not only simplifies the process of finding solutions but also provides deeper insights into the structure and behavior of quadratic expressions.

Finding the Solutions

Having factored the quadratic equation into $(x + 5)(x + 12) = 0$ , we are now in a position to find the solutions for $x$ . The solutions, also known as roots, are the values of $x$ that satisfy the equation, making the entire expression equal to zero. In this factored form, we can 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 property is the cornerstone of solving equations by factoring and provides a direct pathway to finding the solutions. By setting each factor equal to zero and solving the resulting simple equations, we can determine the values of $x$ that make the quadratic equation true. This method is not only efficient but also conceptually clear, making it a valuable tool in any mathematical problem-solving toolkit.

To find the solutions, we set each factor equal to zero:

$x + 5 = 0$ or $x + 12 = 0$

Solving for $x$ in each case:

For $x + 5 = 0$ , subtract 5 from both sides:

$x = -5$

For $x + 12 = 0$ , subtract 12 from both sides:

$x = -12$

Thus, the solutions of the equation $x^2 + 17x + 60 = 0$ are $x = -5$ and $x = -12$ . These values are the points where the quadratic function intersects the x-axis on a graph. Understanding how to find these solutions is critical for various applications in mathematics and other fields, such as physics and engineering. These solutions not only complete the algebraic process but also provide valuable information about the behavior of the quadratic function, including its intercepts and symmetry. Mastering this step is essential for anyone seeking a deeper understanding of quadratic equations and their applications.

In summary, the equivalent equation in factored form is $(x + 5)(x + 12) = 0$ , and the solutions of the equation $-7x - 60 = x^2 + 10x$ are $x = -5$ and $x = -12$ .