Mastering Quadratic Equations Step-by-Step Solutions

Aug 4, 2025 by Admin 53 views

Solving Quadratic Equations: A Comprehensive Guide

Hey guys! 👋 Today, we're diving deep into the fascinating world of quadratic equations. We'll be tackling a set of problems that will help you master the art of solving these equations. Get ready to sharpen your pencils and put on your thinking caps! We'll break down each equation step-by-step, ensuring you understand not just the how, but also the why behind each solution. Let's get started and make some math magic happen! ✨

Let's Solve Some Quadratic Equations

We have a series of quadratic equations to solve, and we’ll use different methods to find the solutions (also known as roots). Let's jump right in!

(8) Solving $2x^2 + x - 10 = 0$

When we look at the quadratic equation $2x^2 + x - 10 = 0$, the main goal is to find the values of x that make this equation true. In simpler terms, we need to figure out what numbers we can plug in for x so that the left side of the equation equals zero. There are several ways to tackle this, but one of the most common and effective methods is factoring. Factoring involves breaking down the quadratic expression into two binomial expressions, which then helps us identify the solutions. Let's walk through the process step by step.

First, we need to find two numbers that multiply to give us the product of the leading coefficient (2) and the constant term (-10), which is -20. At the same time, these two numbers must add up to the middle coefficient, which is 1. Think of it like a puzzle: we're looking for two pieces that fit perfectly into our equation. After some thought, we can identify that the numbers 5 and -4 fit these criteria perfectly. They multiply to -20 (5 * -4 = -20) and add up to 1 (5 + -4 = 1).

Now that we've found these numbers, we can rewrite the middle term of our quadratic equation using these values. Instead of '+ x', we'll write '+ 5x - 4x'. This might seem a bit odd at first, but it's a crucial step in the factoring process. Our equation now looks like this: $2x^2 + 5x - 4x - 10 = 0$.

Next, we'll use a technique called factoring by grouping. We'll group the first two terms together and the last two terms together and then factor out the greatest common factor (GCF) from each group. From the first group, $2x^2 + 5x$, the GCF is x. Factoring out x gives us $x(2x + 5)$. From the second group, $-4x - 10$, the GCF is -2. Factoring out -2 gives us $-2(2x + 5)$. So, our equation now looks like this: $x(2x + 5) - 2(2x + 5) = 0$.

Notice something cool? The term $(2x + 5)$ appears in both parts of our expression. This is exactly what we want! We can factor out this common binomial, which gives us $(x - 2)(2x + 5) = 0$. Now we have our original quadratic expression factored into two binomials.

To find the solutions for x, we set each factor equal to zero. This is because if the product of two factors is zero, then at least one of the factors must be zero. So, we have two equations to solve: $x - 2 = 0$ and $2x + 5 = 0$.

Solving the first equation, $x - 2 = 0$, is simple: just add 2 to both sides, and we get $x = 2$.

Solving the second equation, $2x + 5 = 0$, involves two steps. First, subtract 5 from both sides, which gives us $2x = -5$. Then, divide both sides by 2 to isolate x, which gives us $x = -rac{5}{2}$.

So, the solutions to the quadratic equation $2x^2 + x - 10 = 0$ are $x = 2$ and $x = -rac{5}{2}$. These are the two values of x that make the equation true. We've successfully navigated this quadratic equation using the power of factoring!

(9) Tackling $3x^2 - 7x - 6 = 0$

Let's move on to the next quadratic equation: $3x^2 - 7x - 6 = 0$. Just like before, our mission is to find the values of x that satisfy this equation. We'll stick with our trusty method of factoring, as it's a reliable way to crack these problems. Get ready to follow along as we break this down step by step.

The first thing we need to do is identify the key numbers for factoring. We need to find two numbers that multiply to give us the product of the leading coefficient (3) and the constant term (-6), which is -18. These same two numbers must also add up to the middle coefficient, which is -7. It’s like a detective game where we’re searching for the perfect clues to unlock the solution.

After a bit of brainstorming, we can see that the numbers -9 and 2 fit the bill perfectly. They multiply to -18 (-9 * 2 = -18) and add up to -7 (-9 + 2 = -7). These are our magic numbers that will help us rewrite the equation in a factorable form.

Now, we rewrite the middle term of our quadratic equation using these numbers. Instead of '- 7x', we'll write '- 9x + 2x'. This might seem like a sneaky trick, but it’s a standard technique in factoring. Our equation now transforms into: $3x^2 - 9x + 2x - 6 = 0$.

Next, we'll apply the factoring by grouping method, just like we did in the previous equation. We group the first two terms together and the last two terms together and then factor out the greatest common factor (GCF) from each group. From the first group, $3x^2 - 9x$, the GCF is $3x$. Factoring out $3x$ gives us $3x(x - 3)$. From the second group, $2x - 6$, the GCF is 2. Factoring out 2 gives us $2(x - 3)$. Our equation now looks like this: $3x(x - 3) + 2(x - 3) = 0$.

Notice the common binomial factor? It's $(x - 3)$, which appears in both parts of our expression. This is a great sign because it means we're on the right track. We can now factor out this common binomial, which gives us $(3x + 2)(x - 3) = 0$. We’ve successfully factored our quadratic expression into two binomials!

To find the solutions for x, we set each factor equal to zero. Remember, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two equations to solve: $3x + 2 = 0$ and $x - 3 = 0$.

Solving the first equation, $3x + 2 = 0$, involves a couple of steps. First, subtract 2 from both sides, which gives us $3x = -2$. Then, divide both sides by 3 to isolate x, which gives us $x = -rac{2}{3}$.

Solving the second equation, $x - 3 = 0$, is straightforward: simply add 3 to both sides, and we get $x = 3$.

So, the solutions to the quadratic equation $3x^2 - 7x - 6 = 0$ are $x = -rac{2}{3}$ and $x = 3$. These are the two values of x that make the equation true. We've conquered another quadratic equation using the power of factoring!

(10) Unraveling $4x^2 + 3x - 7 = 0$

Alright, let's keep the momentum going and tackle the next equation: $4x^2 + 3x - 7 = 0$. Our goal remains the same: find the values of x that satisfy this equation. We'll continue using factoring, as it’s a tried-and-true method for solving quadratic equations. Let’s dive in and see how we can break this down.

As with the previous equations, we start by identifying the key numbers for factoring. We need to find two numbers that multiply to give us the product of the leading coefficient (4) and the constant term (-7), which is -28. These same two numbers must also add up to the middle coefficient, which is 3. Think of it as a numerical puzzle where we need to find the right pieces to fit together.

After a bit of thought and calculation, we can see that the numbers 7 and -4 fit the criteria perfectly. They multiply to -28 (7 * -4 = -28) and add up to 3 (7 + -4 = 3). These are the numbers we’ll use to rewrite our equation.

Now, we rewrite the middle term of our quadratic equation using these numbers. Instead of '+ 3x', we'll write '+ 7x - 4x'. This step is crucial for setting up the equation for factoring by grouping. Our equation now becomes: $4x^2 + 7x - 4x - 7 = 0$.

Next, we apply the factoring by grouping method. We group the first two terms together and the last two terms together and then factor out the greatest common factor (GCF) from each group. From the first group, $4x^2 + 7x$, the GCF is x. Factoring out x gives us $x(4x + 7)$. From the second group, $-4x - 7$, the GCF is -1. Factoring out -1 gives us $-1(4x + 7)$. So, our equation now looks like this: $x(4x + 7) - 1(4x + 7) = 0$.

Notice the common binomial factor? It's $(4x + 7)$, which appears in both parts of our expression. This is a clear indication that we're on the right path. We can now factor out this common binomial, which gives us $(x - 1)(4x + 7) = 0$. We’ve successfully factored our quadratic expression into two binomials!

Solving the first equation, $x - 1 = 0$, is simple: just add 1 to both sides, and we get $x = 1$.

Solving the second equation, $4x + 7 = 0$, involves a couple of steps. First, subtract 7 from both sides, which gives us $4x = -7$. Then, divide both sides by 4 to isolate x, which gives us $x = -rac{7}{4}$.

So, the solutions to the quadratic equation $4x^2 + 3x - 7 = 0$ are $x = 1$ and $x = -rac{7}{4}$. These are the two values of x that make the equation true. We’ve successfully unraveled another quadratic equation using factoring!

(11) Cracking $5x^2 - 14x + 8 = 0$

Let's keep going and tackle another quadratic equation: $5x^2 - 14x + 8 = 0$. Our mission remains the same: to find the values of x that make this equation true. We'll stick with our reliable method of factoring to solve this one. Let's get started and see how we can break this down step by step.

The first thing we need to do is identify the key numbers for factoring. We're looking for two numbers that multiply to give us the product of the leading coefficient (5) and the constant term (8), which is 40. These same two numbers must also add up to the middle coefficient, which is -14. It's like a puzzle where we need to find the perfect pieces that fit together to solve it.

After some careful consideration, we can see that the numbers -10 and -4 fit the criteria perfectly. They multiply to 40 (-10 * -4 = 40) and add up to -14 (-10 + -4 = -14). These are the numbers we'll use to rewrite our equation in a factorable form.

Now, we rewrite the middle term of our quadratic equation using these numbers. Instead of '- 14x', we'll write '- 10x - 4x'. This is a crucial step in the factoring process. Our equation now becomes: $5x^2 - 10x - 4x + 8 = 0$.

Next, we'll use the factoring by grouping method, just like we've done before. We group the first two terms together and the last two terms together and then factor out the greatest common factor (GCF) from each group. From the first group, $5x^2 - 10x$, the GCF is $5x$. Factoring out $5x$ gives us $5x(x - 2)$. From the second group, $-4x + 8$, the GCF is -4. Factoring out -4 gives us $-4(x - 2)$. Our equation now looks like this: $5x(x - 2) - 4(x - 2) = 0$.

Notice the common binomial factor? It's $(x - 2)$, which appears in both parts of our expression. This is a great sign because it means we're on the right track. We can now factor out this common binomial, which gives us $(5x - 4)(x - 2) = 0$. We’ve successfully factored our quadratic expression into two binomials!

Solving the first equation, $5x - 4 = 0$, involves a couple of steps. First, add 4 to both sides, which gives us $5x = 4$. Then, divide both sides by 5 to isolate x, which gives us $x = rac{4}{5}$.

Solving the second equation, $x - 2 = 0$, is straightforward: simply add 2 to both sides, and we get $x = 2$.

So, the solutions to the quadratic equation $5x^2 - 14x + 8 = 0$ are $x = rac{4}{5}$ and $x = 2$. These are the two values of x that make the equation true. We've cracked another quadratic equation using the power of factoring!

(12) Decoding $6x^2 + 7x - 20 = 0$

Let’s keep the ball rolling and move on to the next equation: $6x^2 + 7x - 20 = 0$. Our goal remains consistent: we need to find the values of x that satisfy this equation. We’ll continue using our trusted method of factoring, as it has proven to be effective. Let’s dive in and see how we can decode this one.

First, we need to identify the key numbers for factoring. We're looking for two numbers that multiply to give us the product of the leading coefficient (6) and the constant term (-20), which is -120. These same two numbers must also add up to the middle coefficient, which is 7. Think of it as a numerical puzzle where we need to find the right pieces that fit perfectly.

After some thought and calculation, we can see that the numbers 15 and -8 fit the criteria perfectly. They multiply to -120 (15 * -8 = -120) and add up to 7 (15 + -8 = 7). These are the numbers we’ll use to rewrite our equation.

Now, we rewrite the middle term of our quadratic equation using these numbers. Instead of '+ 7x', we'll write '+ 15x - 8x'. This step is crucial for setting up the equation for factoring by grouping. Our equation now becomes: $6x^2 + 15x - 8x - 20 = 0$.

Next, we apply the factoring by grouping method. We group the first two terms together and the last two terms together and then factor out the greatest common factor (GCF) from each group. From the first group, $6x^2 + 15x$, the GCF is $3x$. Factoring out $3x$ gives us $3x(2x + 5)$. From the second group, $-8x - 20$, the GCF is -4. Factoring out -4 gives us $-4(2x + 5)$. So, our equation now looks like this: $3x(2x + 5) - 4(2x + 5) = 0$.

Notice the common binomial factor? It's $(2x + 5)$, which appears in both parts of our expression. This is a clear indication that we're on the right path. We can now factor out this common binomial, which gives us $(3x - 4)(2x + 5) = 0$. We’ve successfully factored our quadratic expression into two binomials!

Solving the first equation, $3x - 4 = 0$, involves a couple of steps. First, add 4 to both sides, which gives us $3x = 4$. Then, divide both sides by 3 to isolate x, which gives us $x = rac{4}{3}$.

Solving the second equation, $2x + 5 = 0$, also involves a couple of steps. First, subtract 5 from both sides, which gives us $2x = -5$. Then, divide both sides by 2 to isolate x, which gives us $x = -rac{5}{2}$.

So, the solutions to the quadratic equation $6x^2 + 7x - 20 = 0$ are $x = rac{4}{3}$ and $x = -rac{5}{2}$. These are the two values of x that make the equation true. We’ve successfully decoded another quadratic equation using factoring!

(13) Deciphering $8x^2 - 6x - 27 = 0$

Alright, let’s keep pushing forward and tackle another quadratic equation: $8x^2 - 6x - 27 = 0$. As always, our goal is to find the values of x that satisfy this equation. We’ll continue to rely on our trusty method of factoring, which has served us well so far. Let’s jump in and see how we can decipher this equation.

Our first step is to identify the key numbers for factoring. We’re looking for two numbers that multiply to give us the product of the leading coefficient (8) and the constant term (-27), which is -216. These same two numbers must also add up to the middle coefficient, which is -6. Think of it like a detective game, where we’re searching for the perfect clues to solve the mystery.

After some careful thought and calculation, we can see that the numbers 12 and -18 fit the criteria perfectly. They multiply to -216 (12 * -18 = -216) and add up to -6 (12 + -18 = -6). These are the numbers we’ll use to rewrite our equation.

Now, we rewrite the middle term of our quadratic equation using these numbers. Instead of '- 6x', we'll write '+ 12x - 18x'. This step is crucial for setting up the equation for factoring by grouping. Our equation now becomes: $8x^2 + 12x - 18x - 27 = 0$.

Next, we apply the factoring by grouping method. We group the first two terms together and the last two terms together and then factor out the greatest common factor (GCF) from each group. From the first group, $8x^2 + 12x$, the GCF is $4x$. Factoring out $4x$ gives us $4x(2x + 3)$. From the second group, $-18x - 27$, the GCF is -9. Factoring out -9 gives us $-9(2x + 3)$. So, our equation now looks like this: $4x(2x + 3) - 9(2x + 3) = 0$.

Notice the common binomial factor? It's $(2x + 3)$, which appears in both parts of our expression. This is a clear sign that we're on the right path. We can now factor out this common binomial, which gives us $(4x - 9)(2x + 3) = 0$. We’ve successfully factored our quadratic expression into two binomials!

Solving the first equation, $4x - 9 = 0$, involves a couple of steps. First, add 9 to both sides, which gives us $4x = 9$. Then, divide both sides by 4 to isolate x, which gives us $x = rac{9}{4}$.

Solving the second equation, $2x + 3 = 0$, also involves a couple of steps. First, subtract 3 from both sides, which gives us $2x = -3$. Then, divide both sides by 2 to isolate x, which gives us $x = -rac{3}{2}$.

So, the solutions to the quadratic equation $8x^2 - 6x - 27 = 0$ are $x = rac{9}{4}$ and $x = -rac{3}{2}$. These are the two values of x that make the equation true. We’ve successfully deciphered another quadratic equation using factoring!

(14) Cracking the Code of $10x^2 - x - 21 = 0$

Last but not least, let's tackle the final quadratic equation in our list: $10x^2 - x - 21 = 0$. Our mission, as always, is to find the values of x that satisfy this equation. We'll stick with our reliable method of factoring to crack this code. Let's dive in and see how we can break it down step by step.

First, we need to identify the key numbers for factoring. We're looking for two numbers that multiply to give us the product of the leading coefficient (10) and the constant term (-21), which is -210. These same two numbers must also add up to the middle coefficient, which is -1. Think of it like a puzzle where we need to find the perfect pieces to fit together and reveal the solution.

After some careful thought and calculation, we can see that the numbers -15 and 14 fit the criteria perfectly. They multiply to -210 (-15 * 14 = -210) and add up to -1 (-15 + 14 = -1). These are the numbers we'll use to rewrite our equation in a factorable form.

Now, we rewrite the middle term of our quadratic equation using these numbers. Instead of '- x', we'll write '- 15x + 14x'. This is a crucial step in the factoring process. Our equation now becomes: $10x^2 - 15x + 14x - 21 = 0$.

Next, we'll use the factoring by grouping method, just like we've done in the previous equations. We group the first two terms together and the last two terms together and then factor out the greatest common factor (GCF) from each group. From the first group, $10x^2 - 15x$, the GCF is $5x$. Factoring out $5x$ gives us $5x(2x - 3)$. From the second group, $14x - 21$, the GCF is 7. Factoring out 7 gives us $7(2x - 3)$. Our equation now looks like this: $5x(2x - 3) + 7(2x - 3) = 0$.

Notice the common binomial factor? It's $(2x - 3)$, which appears in both parts of our expression. This is a great sign because it means we're on the right track. We can now factor out this common binomial, which gives us $(5x + 7)(2x - 3) = 0$. We’ve successfully factored our quadratic expression into two binomials!

Solving the first equation, $5x + 7 = 0$, involves a couple of steps. First, subtract 7 from both sides, which gives us $5x = -7$. Then, divide both sides by 5 to isolate x, which gives us $x = -rac{7}{5}$.

Solving the second equation, $2x - 3 = 0$, also involves a couple of steps. First, add 3 to both sides, which gives us $2x = 3$. Then, divide both sides by 2 to isolate x, which gives us $x = rac{3}{2}$.

So, the solutions to the quadratic equation $10x^2 - x - 21 = 0$ are $x = -rac{7}{5}$ and $x = rac{3}{2}$. These are the two values of x that make the equation true. We've cracked the code of this quadratic equation using factoring!

Wrapping Up

And there you have it, folks! We've successfully solved a series of quadratic equations using the method of factoring. Remember, practice makes perfect, so keep honing your skills, and you'll become a quadratic equation-solving pro in no time! If you ever get stuck, just break it down step by step, and you'll get there. Keep up the fantastic work! 🎉