Solving Quadratic Equations And Inequalities A Step-by-Step Guide

Hey guys! Let's dive into the world of quadratic equations and inequalities. This guide will walk you through solving for 'x' in different scenarios, from completing the square to handling inequalities. We'll break it down step by step, making sure you grasp each concept. So, grab your pencils, and let's get started!

Solving 2x - 14 = -rac{20}{x} by Completing the Square

When faced with an equation like 2x - 14 = -rac{20}{x}, our goal is to isolate 'x'. This particular equation involves a fraction, which adds a little twist, but don't worry, we'll tackle it together. The method we'll use here is completing the square, a technique that's super useful for solving quadratic equations. This method transforms the quadratic equation into a perfect square trinomial, making it easier to solve.

Step 1: Clear the Fraction

Fractions can be a bit annoying to work with, so let's get rid of it first. We have a term with 'x' in the denominator, specifically $-\frac{20}{x}$. To eliminate this fraction, we'll multiply every term in the equation by 'x'. This gives us:

$x(2x - 14) = x(-\frac{20}{x})$

Simplifying this, we get:

$2x^2 - 14x = -20$

Now, our equation looks much cleaner and easier to handle. We've successfully transformed the original equation into a more manageable form by clearing the fraction. This is a crucial first step in solving equations involving rational expressions. By multiplying each term by 'x', we ensure that the equation remains balanced while eliminating the denominator.

Step 2: Rearrange into Standard Quadratic Form

To apply the method of completing the square, we need our equation in the standard quadratic form, which is $ax^2 + bx + c = 0$. Currently, our equation is $2x^2 - 14x = -20$. To get it into the standard form, we need to move the constant term (-20) to the left side of the equation. We can do this by adding 20 to both sides:

$2x^2 - 14x + 20 = -20 + 20$

This simplifies to:

$2x^2 - 14x + 20 = 0$

Now, our equation is in the standard quadratic form, making it ready for the next steps in completing the square. This rearrangement is essential because the standard form allows us to easily identify the coefficients a, b, and c, which are crucial for applying the completing the square technique. By adding 20 to both sides, we ensure that the equation remains balanced while achieving the desired form.

Step 3: Divide by the Leading Coefficient

Completing the square works best when the coefficient of the $x^2$ term (the 'a' value) is 1. In our equation, $2x^2 - 14x + 20 = 0$, the leading coefficient is 2. To make it 1, we'll divide every term in the equation by 2:

$\frac{2x^2}{2} - \frac{14x}{2} + \frac{20}{2} = \frac{0}{2}$

This simplifies to:

$x^2 - 7x + 10 = 0$

Now, the coefficient of the $x^2$ term is 1, which sets us up perfectly for completing the square. Dividing by the leading coefficient is a critical step because it simplifies the process of finding the constant term needed to complete the square. This step ensures that we are working with a simpler quadratic expression, making the subsequent calculations more manageable and accurate.

Step 4: Complete the Square

This is the heart of the method! We need to add and subtract a value to the left side of the equation that will turn the $x^2 - 7x$ part into a perfect square trinomial. Here's how we find that value:

1. Take the coefficient of the x term (-7), divide it by 2 (giving us -\frac{7}{2}), and then square the result.

$(- \frac{7}{2})^2 = \frac{49}{4}$

2. Now, we add and subtract this value within the equation:

$x^2 - 7x + \frac{49}{4} - \frac{49}{4} + 10 = 0$

The first three terms, $x^2 - 7x + \frac{49}{4}$, now form a perfect square trinomial. We can rewrite it as:

$(x - \frac{7}{2})^2$

So, our equation becomes:

$(x - \frac{7}{2})^2 - \frac{49}{4} + 10 = 0$

Completing the square involves transforming a quadratic expression into a perfect square trinomial plus a constant. This is achieved by adding and subtracting a specific value, derived from the coefficient of the x term, which allows us to rewrite the quadratic expression in a form that is easily factored as a square. This step is fundamental to the completing the square method, enabling us to solve for x by isolating the squared term.

Step 5: Simplify and Isolate the Squared Term

Let's simplify the equation $(x - \frac{7}{2})^2 - \frac{49}{4} + 10 = 0$. We need to combine the constant terms. To do this, we'll find a common denominator for $-\frac{49}{4}$ and 10. The common denominator is 4, so we rewrite 10 as $\frac{40}{4}$:

$(x - \frac{7}{2})^2 - \frac{49}{4} + \frac{40}{4} = 0$

Now, combine the fractions:

$(x - \frac{7}{2})^2 - \frac{9}{4} = 0$

Next, we want to isolate the squared term. To do this, we'll add $\frac{9}{4}$ to both sides of the equation:

$(x - \frac{7}{2})^2 = \frac{9}{4}$

We've now successfully isolated the squared term, which is a crucial step in solving for x. This simplification allows us to easily apply the square root property in the next step. By isolating the squared term, we set the stage for directly solving for the variable, making the solution process more straightforward and efficient.

Step 6: Take the Square Root

Now that we have $(x - \frac{7}{2})^2 = \frac{9}{4}$, we can take the square root of both sides. Remember, when we take the square root, we need to consider both the positive and negative roots:

$\sqrt{(x - \frac{7}{2})^2} = \pm \sqrt{\frac{9}{4}}$

This simplifies to:

$x - \frac{7}{2} = \pm \frac{3}{2}$

Taking the square root of both sides is a key step in solving equations where a variable is squared. By considering both the positive and negative roots, we ensure that we capture all possible solutions for x. This step directly follows from isolating the squared term and allows us to move closer to finding the values of x that satisfy the original equation.

Step 7: Solve for $x$

We now have two separate equations to solve:

1. $x - \frac{7}{2} = \frac{3}{2}$
2. $x - \frac{7}{2} = -\frac{3}{2}$

Let's solve the first equation. Add $\frac{7}{2}$ to both sides:

$x = \frac{3}{2} + \frac{7}{2}$

$x = \frac{10}{2}$

$x = 5$

Now, let's solve the second equation. Add $\frac{7}{2}$ to both sides:

$x = -\frac{3}{2} + \frac{7}{2}$

$x = \frac{4}{2}$

$x = 2$

So, the solutions for $x$ are 5 and 2. Solving for x involves isolating x in each of the equations derived from taking the square root. By adding the necessary terms to both sides, we find the specific values of x that satisfy the original equation. This final step is crucial for determining the solutions to the quadratic equation and completing the problem.

Final Answer

The solutions for $x$ are 5 and 2. We found these by systematically clearing fractions, rearranging the equation into standard quadratic form, completing the square, and then isolating and solving for $x$. Each step is crucial in ensuring we arrive at the correct solutions. Remember to always check your solutions by plugging them back into the original equation to verify their correctness.

Solving the Inequality $-x^2 + 9x - 20 \\\leq 0$

Next, let's tackle the inequality $-x^2 + 9x - 20 \\leq 0$. Solving inequalities involves finding the range of values for 'x' that satisfy the given condition. This particular inequality is quadratic, so we'll use a combination of factoring and sign analysis to find the solution set. Inequalities differ from equations in that they represent a range of values rather than specific points, which requires a different approach to solve.

Step 1: Factor the Quadratic Expression

To solve the inequality, we first need to factor the quadratic expression $-x^2 + 9x - 20$. Factoring helps us find the critical points where the expression equals zero, which are essential for determining the intervals where the inequality holds true. Let's factor the expression:

First, it's easier to work with a positive leading coefficient, so let's factor out a -1:

$-1(x^2 - 9x + 20) \\leq 0$

Now, we need to factor the quadratic expression $x^2 - 9x + 20$. We're looking for two numbers that multiply to 20 and add up to -9. Those numbers are -4 and -5.

So, we can factor the expression as:

$-1(x - 4)(x - 5) \\leq 0$

Factoring the quadratic expression is a critical step because it allows us to identify the roots of the equation, which are the points where the expression changes sign. These roots divide the number line into intervals that we will analyze to determine where the inequality is satisfied. Factoring simplifies the inequality into a product of linear factors, making it easier to work with.

Step 2: Identify Critical Points

The critical points are the values of $x$ that make the factored expression equal to zero. These points divide the number line into intervals where the expression will be either positive or negative. From our factored inequality, $-1(x - 4)(x - 5) \\leq 0$, the critical points are the values of $x$ that make each factor zero:

$x - 4 = 0 \\implies x = 4$

$x - 5 = 0 \\implies x = 5$

So, the critical points are $x = 4$ and $x = 5$. These points are crucial because they are the boundaries where the expression can change its sign. Identifying the critical points is a key step in solving inequalities, as it sets the stage for analyzing the intervals between these points to determine where the inequality holds true. The critical points essentially segment the number line, allowing us to examine each segment separately.

Step 3: Create a Sign Chart

A sign chart helps us visualize the sign of the expression $-1(x - 4)(x - 5)$ in each interval determined by the critical points. We'll create a number line and mark the critical points, then test a value from each interval to determine the sign of the expression in that interval. This is a systematic way to analyze how the sign of the expression changes across different intervals.

1. Draw a number line and mark the critical points 4 and 5.

2. Choose a test value from each interval: $x < 4$, $4 < x < 5$, and $x > 5$.

   • For $x < 4$, let's choose $x = 3$:

     $-1(3 - 4)(3 - 5) = -1(-1)(-2) = -2$ (negative)

   • For $4 < x < 5$, let's choose $x = 4.5$:

     $-1(4.5 - 4)(4.5 - 5) = -1(0.5)(-0.5) = 0.25$ (positive)

   • For $x > 5$, let's choose $x = 6$:

     $-1(6 - 4)(6 - 5) = -1(2)(1) = -2$ (negative)

3. Create the sign chart:

   Interval  | x < 4 | 4 < x < 5 | x > 5 |
   ----------|-------|-----------|-------|
   x - 4     |   -   |     +     |   +   |
   x - 5     |   -   |     -     |   +   |
   -1        |   -   |     -     |   -   |
   Expression|   -   |     +     |   -   |
   

The sign chart provides a clear visual representation of how the sign of the expression changes across different intervals. This tool is essential for determining the intervals where the inequality is satisfied. By testing values within each interval, we can confidently assess the sign of the expression, making it easier to identify the solution set for the inequality.

Step 4: Determine the Solution Set

We want to find the values of $x$ where $-1(x - 4)(x - 5) \\leq 0$. From the sign chart, we see that the expression is negative when $x < 4$ and when $x > 5$. Since the inequality includes “equal to” (\leq), we also include the critical points where the expression equals zero.

So, the solution set is $x \\leq 4$ or $x \\geq 5$. This can be written in interval notation as $(-\\infty, 4] \\cup [5, \\infty)$. The solution set represents all values of x that satisfy the original inequality. By including the critical points, we ensure that all values where the expression is either negative or zero are part of the solution.

Final Answer

The solution set for the inequality $-x^2 + 9x - 20 \\leq 0$ is $x \\leq 4$ or $x \\geq 5$, which in interval notation is $(-\\infty, 4] \\cup [5, \\infty)$. We arrived at this solution by factoring the quadratic expression, identifying the critical points, creating a sign chart, and then determining the intervals where the expression is negative or zero. This process provides a systematic way to solve quadratic inequalities.

Solving a System of Equations: $3y = 3x - 6$ and $y = (x + 2)(x - 3) + 4$

Now, let's move on to solving a system of equations. We have two equations:

1. $3y = 3x - 6$
2. $y = (x + 2)(x - 3) + 4$

To solve this system, we need to find the values of $x$ and $y$ that satisfy both equations simultaneously. We'll use the method of substitution, which involves solving one equation for one variable and then substituting that expression into the other equation. This method is effective for systems where one equation can be easily solved for one variable, making it a straightforward approach to finding the solutions.

Step 1: Simplify the First Equation

The first equation, $3y = 3x - 6$, can be simplified by dividing every term by 3:

$\frac{3y}{3} = \frac{3x}{3} - \frac{6}{3}$

This simplifies to:

$y = x - 2$

Now, we have a simpler expression for $y$ in terms of $x$. Simplifying the first equation makes it easier to substitute into the second equation. By dividing each term by 3, we isolate y and express it directly in terms of x, which sets us up nicely for the substitution method.

Step 2: Substitute into the Second Equation

We'll substitute the expression for $y$ from the simplified first equation ($y = x - 2$) into the second equation, $y = (x + 2)(x - 3) + 4$:

$x - 2 = (x + 2)(x - 3) + 4$

This substitution replaces $y$ in the second equation with an expression involving only $x$, allowing us to solve for $x$. The substitution method is a powerful technique for solving systems of equations because it reduces the system to a single equation with one variable, making it easier to find the solution.

Step 3: Expand and Simplify

Now, let's expand and simplify the equation $x - 2 = (x + 2)(x - 3) + 4$. First, expand the product $(x + 2)(x - 3)$:

$(x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$

So, our equation becomes:

$x - 2 = x^2 - x - 6 + 4$

Now, combine the constants:

$x - 2 = x^2 - x - 2$

Expanding and simplifying the equation is a crucial step in solving for x. By removing the parentheses and combining like terms, we transform the equation into a more manageable form, specifically a quadratic equation. This simplification is necessary to apply standard techniques for solving quadratic equations, such as setting the equation to zero and factoring.

Step 4: Rearrange into Standard Quadratic Form

To solve for $x$, we need to rearrange the equation $x - 2 = x^2 - x - 2$ into the standard quadratic form, $ax^2 + bx + c = 0$. Let's move all the terms to one side of the equation:

$0 = x^2 - x - 2 - x + 2$

Combine like terms:

$0 = x^2 - 2x$

Rearranging the equation into standard quadratic form is essential because it allows us to use methods like factoring or the quadratic formula to solve for x. By setting one side of the equation to zero, we create a format that is conducive to these solution techniques. This step is a standard practice in solving quadratic equations and is critical for finding the values of x that satisfy the equation.

Step 5: Factor and Solve for $x$

Now, let's factor the quadratic equation $0 = x^2 - 2x$. We can factor out an $x$:

$0 = x(x - 2)$

Now, we set each factor equal to zero and solve for $x$:

$x = 0$

$x - 2 = 0 \\implies x = 2$

So, the possible values for $x$ are 0 and 2. Factoring the quadratic equation is a straightforward way to find the roots, which are the values of x that make the equation equal to zero. By setting each factor to zero, we obtain the possible solutions for x. This step is a direct application of 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.

Step 6: Substitute $x$ Values to Find $y$

Now that we have the values for $x$, we need to find the corresponding values for $y$. We'll use the simplified first equation, $y = x - 2$, to find $y$ for each value of $x$:

1. For $x = 0$:

   $y = 0 - 2 = -2$

2. For $x = 2$:

   $y = 2 - 2 = 0$

So, the solutions are $(0, -2)$ and $(2, 0)$. Substituting the x values back into one of the original equations allows us to find the corresponding y values. This step is crucial for completing the solution to the system of equations, as it provides the ordered pairs (x, y) that satisfy both equations simultaneously. By using the simplified equation, we make this step more efficient and straightforward.

Final Answer

The solutions to the system of equations are $(0, -2)$ and $(2, 0)$. We found these solutions by simplifying one equation, substituting the expression into the other equation, expanding and simplifying, rearranging into standard quadratic form, factoring to solve for $x$, and then substituting the $x$ values back to find the corresponding $y$ values. This systematic approach ensures we find all possible solutions for the system.

Conclusion

Alright, guys, we've covered a lot! From completing the square to solving inequalities and systems of equations, you've now got a solid toolkit for tackling quadratic problems. Remember, practice makes perfect, so keep working through examples, and you'll become a quadratic equation master in no time! Solving quadratic equations and inequalities is a fundamental skill in mathematics, and mastering these techniques opens the door to more advanced topics and applications. Keep up the great work!