Solving The Quadratic Equation -7x-60=x^2+10x

Understanding quadratic equations is crucial for anyone delving into algebra and beyond. This article aims to provide a comprehensive guide on how to solve a specific quadratic equation, $-7x - 60 = x^2 + 10x$, by transforming it into an equivalent form and finding its solutions. We will walk through the steps systematically, ensuring a clear understanding of the underlying principles and techniques.

Transforming the Equation into Standard Form

The first step in solving any quadratic equation is to rewrite it in the standard form, which is $ax^2 + bx + c = 0$. This form allows us to easily identify the coefficients $a$, $b$, and $c$, which are essential for applying various solution methods such as factoring, completing the square, or using the quadratic formula. Starting with the given equation, $-7x - 60 = x^2 + 10x$, we need to rearrange the terms to bring it into the standard form. To do this, we will add $7x$ and $60$ to both sides of the equation. This ensures that we maintain the equality while moving all terms to one side.

Adding $7x$ to both sides gives us $-60 = x^2 + 17x$. Next, adding $60$ to both sides results in $0 = x^2 + 17x + 60$. Now, we have the quadratic equation in the standard form, $x^2 + 17x + 60 = 0$. In this form, it is clear that $a = 1$, $b = 17$, and $c = 60$. These coefficients will be vital as we proceed to solve the equation. Recognizing the standard form is a foundational skill, as it sets the stage for choosing the most appropriate method to find the solutions. This transformation not only simplifies the equation but also aligns it with established mathematical practices, making it easier to apply subsequent steps.

Factoring the Quadratic Equation

Now that we have the quadratic equation in the standard form, $x^2 + 17x + 60 = 0$, the next step is to factor it. Factoring involves expressing the quadratic expression as a product of two binomials. This method is particularly effective when the quadratic equation has integer solutions, which makes it a quick and straightforward approach. To factor the equation, we need to find two numbers that multiply to the constant term ($c = 60$) and add up to the coefficient of the linear term ($b = 17$). This is a critical step, as the correct factors will lead us to the solutions of the equation. After some consideration, we can identify that the two numbers are $12$ and $5$ because $12 imes 5 = 60$ and $12 + 5 = 17$.

Using these numbers, we can rewrite the middle term of the quadratic equation as a sum of two terms: $x^2 + 12x + 5x + 60 = 0$. Now, we can factor by grouping. We group the first two terms and the last two terms: $(x^2 + 12x) + (5x + 60) = 0$. Next, we factor out the greatest common factor (GCF) from each group. From the first group, the GCF is $x$, and from the second group, the GCF is $5$. Factoring these out, we get $x(x + 12) + 5(x + 12) = 0$. Notice that both terms now have a common factor of $(x + 12)$. We can factor this out, resulting in $(x + 12)(x + 5) = 0$. This is the factored form of the quadratic equation, and it is equivalent to the original equation. Factoring is a powerful technique that simplifies the process of finding solutions by breaking down the quadratic expression into manageable parts.

Determining the Solutions

With the quadratic equation factored as $(x + 12)(x + 5) = 0$, we can now easily find the solutions. The principle we use here is 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. In other words, if $AB = 0$, then either $A = 0$ or $B = 0$ (or both). Applying this property to our factored equation, we set each factor equal to zero and solve for $x$.

First, we set $(x + 12) = 0$. Subtracting $12$ from both sides gives us $x = -12$. This is one solution of the quadratic equation. Next, we set $(x + 5) = 0$. Subtracting $5$ from both sides gives us $x = -5$. This is the second solution of the quadratic equation. Therefore, the solutions to the equation $x^2 + 17x + 60 = 0$ are $x = -12$ and $x = -5$. These values of $x$ are the points where the quadratic function intersects the x-axis on a graph. Finding these solutions is the ultimate goal of solving the equation, and the factored form makes this process straightforward. In summary, by factoring the quadratic equation and applying the zero-product property, we efficiently determined the two solutions, providing a clear and concise answer to the problem.

Writing the Equivalent Equation

Based on the factoring process we went through, the equivalent equation in the form (x + ext{_})(x + ext{_}) = 0 is $(x + 12)(x + 5) = 0$.

Solutions of the Quadratic Equation

The solutions of the equation $-7x - 60 = x^2 + 10x$ are $x = -12$ and $x = -5$.

The solutions of the equation $-7x - 60 = x^2 + 10x$ are the values of $x$ that make the equation true. To find these solutions, we first need to rearrange the equation into the standard quadratic form, which is $ax^2 + bx + c = 0$. This involves moving all terms to one side of the equation. Starting with $-7x - 60 = x^2 + 10x$, we can add $7x$ and $60$ to both sides to get $0 = x^2 + 10x + 7x + 60$. Combining like terms, we have $0 = x^2 + 17x + 60$. Now, the equation is in the standard quadratic form, with $a = 1$, $b = 17$, and $c = 60$.

Once the equation is in standard form, we can solve it using several methods, such as factoring, completing the square, or using the quadratic formula. Factoring is often the quickest method if the quadratic expression can be easily factored. In this case, we are looking for two numbers that multiply to $60$ (the constant term) and add up to $17$ (the coefficient of the $x$ term). These numbers are $12$ and $5$, since $12 imes 5 = 60$ and $12 + 5 = 17$. Thus, we can factor the quadratic expression as $(x + 12)(x + 5) = 0$. This factored form makes it easy to find the solutions.

To find the solutions, 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. So, we set each factor equal to zero: $x + 12 = 0$ and $x + 5 = 0$. Solving the first equation, $x + 12 = 0$, we subtract $12$ from both sides to get $x = -12$. Solving the second equation, $x + 5 = 0$, we subtract $5$ from both sides to get $x = -5$. Therefore, the solutions of the quadratic equation $-7x - 60 = x^2 + 10x$ are $x = -12$ and $x = -5$. These are the values of $x$ that satisfy the original equation, making the left and right sides equal.

Verifying the Solutions

To ensure our solutions are correct, we can substitute each value back into the original equation and check if it holds true. This is a crucial step in solving any equation, as it helps to catch any errors that may have occurred during the solution process. Let's start by verifying the solution $x = -12$. Substituting $x = -12$ into the original equation, $-7x - 60 = x^2 + 10x$, we get $-7(-12) - 60 = (-12)^2 + 10(-12)$. Simplifying the left side, we have $84 - 60 = 24$. Simplifying the right side, we have $144 - 120 = 24$. Since both sides are equal ($24 = 24$), the solution $x = -12$ is correct.

Next, let's verify the solution $x = -5$. Substituting $x = -5$ into the original equation, $-7x - 60 = x^2 + 10x$, we get $-7(-5) - 60 = (-5)^2 + 10(-5)$. Simplifying the left side, we have $35 - 60 = -25$. Simplifying the right side, we have $25 - 50 = -25$. Since both sides are equal ($-25 = -25$), the solution $x = -5$ is also correct. Therefore, we have verified that both $x = -12$ and $x = -5$ are indeed solutions to the quadratic equation $-7x - 60 = x^2 + 10x$. This verification step confirms our solution and provides confidence in our answer.

In conclusion, by rearranging the equation into standard form, factoring the quadratic expression, applying the zero-product property, and verifying our solutions, we have successfully found and confirmed that the solutions of the equation $-7x - 60 = x^2 + 10x$ are $x = -12$ and $x = -5$. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis.

$x = -12, -5$