Decoding Polynomial Equations: How to Solve Quadratic Equation 4x² – 5x – 12 = 0?
This problem looks at the equation 4x^2 – 5x – 12 = 0. This is called a quadratic equation cause it has x squared being multiplied by 4. Also, quadratic equations are ones where the highest power of the variable x is 2.
This equation has 3 parts – the x squared part, the x part, and the number part at the end. The x-squared part is 4x^2. The x part is -5x – it’s negative cause of the minus sign. And the end part is -12.
We want to solve this equation to find the value of x that makes the two sides equal. To do that, we use a method called factoring. Also, factoring breaks the equation into two pieces multiplied together that each equals 0. Figuring out these factors will tell us the value of x.
So, let’s break this quadratic equation down step-by-step to find its solutions.
What Is a Quadratic Equation Anyway?
A quadratic equation is a special math problem where x is raised to the power of 2. The biggest exponent on x is 2.
Quadratic equations look like this: ax2 + bx + c = 0. The part is multiplied by x squared. Also, the b part multiplies x alone. And the c is a number by itself.
In our equation, 4×2 is the part cause 4 multiplies x squared. -5x is the b part cause -5 goes with x. And -12 is the c part cause that’s just a plain number.
The x2 part is what makes it quadratic. Also, normal equations might have x or a number. But quadratics specifically deal with x squared.
The goal with quadratics is to figure out what number x has to be for the two sides to be equal. We’ll use factoring that breaks it into pieces that each equal 0. Then, we can find the x values.
The Quadratic Equation Concept
A quadratic equation deals with variable x raised to the power of 2. Also, this makes it different from other equations. The variable is squared instead of just itself.
All quadratic equations can fit the formula ax2 + bx + c = 0. In our equation, the a part is 4×2. This means 4 multiplied by x squared. The b part is -5x. So -5 goes with just x. Also, the c is -12, which is a plain number.
This shows the standard form for quadratics with the x squared term first, then x, and then the number. Also, we aim to factor this equation into two parts, each equal to 0 when multiplied.
Factoring is the key to solving quadratics. It breaks the equation apart to find the x values that equal both sides. Once factored, the equation will be something like (x – something) (x – something else) = 0.
Whatever x equals in those factors will satisfy the original equation. By factoring, we decode what x needs to be to make the left and right sides the same. Also, this lets us solve the quadratic problem.
Method of Solution 4x ^ 2 – 5x – 12 = 0
To solve this quadratic equation, we use the method of factoring. Factoring breaks the equation into two pieces that are multiplied together. Also, we look for two numbers that multiply to give the constant term (-12) and add up to give the coefficient of the linear term (-5x).
We can factor this equation as (4x – 3)(x + 4) = 0. Also, we get the numbers -3 and 4 because -3 × 4 is -12, and -3 + 4 is -5.
Now, the equation is in a factored form where each factor equals 0. Also, for the first factor (4x – 3) to equal 0, 4x has to equal 3. So x equals 3/4. For the second factor (x + 4) to equal 0, x has to equal -4.
Therefore, the two solutions or values of x that make the original equation true are x = 3/4 and x = -4. Factoring allows us to break down the equation and more easily see what x needs to be for the left and right sides to be equal.
How Do We Find the Solution?
Here are some solutions are given below:
Calculation
To solve the equation, we use factoring.
- First, we look for two numbers that multiply to the constant term (-12) and add up to the coefficient of the linear term (-5x). Also, the two numbers that work are -3 and 4.
- Next, we rewrite the left side and factor it by grouping the first and last two terms: 4x^2 – 5x – 12 = (4x^2 – 3x) + (-4x – 12).
- Then we factor out the greatest common factor from each group: (4x^2 – 3x) + (-4x – 12) = (4x – 3)(x + 4).
- Setting each factor equal to 0 gives: (4x – 3) = 0 and (x + 4) = 0.
- Solving the first equation, 4x – 3 = 0, gives x = 3/4.
- Solving the second equation, x + 4 = 0, gives x = -4.
Therefore, the two solutions are:
- x = 3/4
- x = -4
We’ve now used factoring to break down and solve the original quadratic equation step-by-step.
Significance and Applications
This problem shows how to solve the quadratic equation 4x² – 5x – 12 = 0 = 0 step-by-step. Also, a quadratic equation has x squared in it. This one follows the standard form of 4x² – 5x – 12 = 0.
To solve it, we use factoring. Factoring breaks the equation into two parts that multiply to give the original equation. For this one, the factors are (4x – 3) and (x + 4).
Setting each factor equal to zero gives us two equations to solve: (4x – 3) = 0 and (x + 4) = 0. Also, solving the first one, 4x – 3 = 0, gives x = 3/4. Solving the second, x + 4 = 0, gives x = -4.
So, the two solutions are x = 3/4 and x = -4. Factoring lets us “decode” the quadratic and find the specific x values that satisfy the equation. Also, this is useful for predicting real-world situations modeled by quadratics, like motion or finances. Being able to solve them is an important math skill.
Real-World Example
Let’s say a basketball player wanted to make a shot from 15 feet away. Also, we can model this with a quadratic equation. Let x = distance to the hoop (in feet). The equation would be:
x^2 – 15x + 100 = 0
This equation takes the player’s initial distance (15 feet), accounts for gravity slowing the ball’s speed over x distance, and sets the target at 100 inches from the floor.
To solve it, we factor:
x^2 – 15x + 100 = (x – 10)(x – 10)
Setting each factor equal to 0 gives:
(x – 10) = 0 and (x – 10) = 0
Solving these, we get x = 10 feet both times.
So, if the player shoots from 10 feet away, the ball will reach a height of 100 inches and swish through the net. Factoring and solving tell us the best shooting distance in this real situation. Also, it shows how math applies to predicting real events.
Uses Of Quadratic Equation (4x² – 5x – 12 = 0)
The quadratic equation is 4x² – 5x – 12 = 0, where a, b, and c are constants. It has various uses in different fields:
Mathematics
Quadratic equations are super useful in math. Also, they help solve many different types of problems. By understanding the quadratic equation, we can graph parabolas and other curved shapes in algebra. This teaches us about functions and relationships.
Calculus involves things that change, like velocity and maximums. Also, we use quadratic equations to differentiate, integrate, and find limits of changing amounts. Statistics models trends over time with techniques like regression. Also, these often involve factoring quadratics to predict patterns from sample data.
Geometry looks at curved shapes like circles and ellipses. Also, the equations that describe these landforms relate to quadratics. Even more advanced math relies on quadratics. We might need to factor one to help solve a system of equations or prove the properties of numbers.
Being comfortable with quadratics opens the door to higher math topics. Also, they appear in algebra, calculus, geometry, statistics, and beyond. Mastering how to solve them is valuable for understanding diverse concepts.
Physics
Quadratic equations show up everywhere! They help solve many types of problems in easy-to-understand ways.
In algebra, we draw curvy parabola shapes by understanding quadratics. Calculus is about changing amounts, like speed and hills. Quadratics help with this.
Statistics looks at how things change over time. Things like surveys use quadratics to see patterns. Geometry looks at round shapes like circles. These are described with quadratic equations, too.
Physics uses quadratics for moving things. Like balls tossed in the air or swinging on a swing. The equations show where things will end up.
Engineering
Engineering designs buildings and machines using quadratics. Economics analyzes supply and demand, which relates to quadratics. Even harder topics sometimes use quadratics to help solve problems or show why things work. Their flexible pattern shows up a lot.
Mastering quadratics lets you understand bigger ideas by practicing this basic math. It helps make sense of different subject areas through one simple lens. Engineers use math a lot in their work. Quadratic equations help with many engineering problems.
Engineers who design structures like buildings, bridges, or machines must know how things will bend or vibrate. Also, these motions follow quadratic patterns. Solving the equations predicts how far things will flex.
Energy is important for engines, circuits, and more. Quadratic formulas track energy changes between stored and working types. Also, this helps engineers make gadgets work better. Engineering also examines circulating water flow, gear interactions, and wheel friction. Also, all exhibit quadratic curves dealt with using these methods.
Whether building tall skyscrapers or efficient motors, quadratic modeling optimizes designs to be strong but lightweight. Also, it minimizes materials and costs. Solving quadratics gives engineers insights into safely constructing structures, machines, and systems. Also, it’s an essential mathematical tool for their important work.
Quadratic Equation Applications About 4x² – 5x – 12 = 0
Here are some applications you need to know:
Projectile Motion
When objects are thrown, shot, or launched into the air, their motion follows a curved path called projectile motion. This type of motion can be described accurately using quadratic equations.
Objects in projectile motion are pulled down by gravity but keep going forward due to their initial speed and the angle at which they were launched. Quadratic formulas model how an object’s height changes over time as it travels through the air.
The quadratic equation determines the initial speed, launch angle, and gravity. By solving the equation, we can find the object’s position – like how high or far it traveled – at any point in its flight.
People use projectile motion equations in different fields. Athletes aim basketball shots, soldiers fire artillery, and scientists launch rockets into space. The math also helps solve physics problems about throws, kicks, and rebounds.
Being able to model real-world projectile motions with quadratics provides important insights. It unlocks an understanding of abstract concepts through practical applications.
Optimal Solutions
You’re at a local fair with a ring toss game. You want to win that giant stuffed bear, don’t you? Here’s where quadratic equations come in handy! They can help find the best solutions for problems.
In this case, they can help you figure out the best way to toss that ring. How? By helping you find the best angle and speed to toss the ring so it lands on the bottle, making you a winner!
Isn’t that awesome? Quadratic equations are like the secret cheat code to win games at the fair!
Electrical Engineering
Did you know the device you’re using to read this blog uses quadratic equations? Cool, huh? Electrical engineers, the folks who design electronic gadgets, use these equations a lot!
With the help of quadratic equations, they design circuits, the tiny paths for electricity to travel inside our devices. Also, these circuits make your computer, phone, or game console work.
Without quadratic equations, you may be unable to play your favorite video games or watch your favorite shows on your tablet. So, next time you’re enjoying your device, remember there’s a quadratic equation behind the scenes!
Architecture and Construction
You know when you’re building a super tall tower with your building blocks and trying to make sure it doesn’t fall over? That’s what real-life architects and construction workers must do but on a much bigger scale!
They use quadratic equations to make sure their buildings are safe. These equations help them determine how much weight a building can handle without toppling over. Also, they help them know how the weight spreads across the building. It’s like when you balance your blocks just right so your tower stays standing.
So, remember, next time you’re playing with your blocks, you’re like a mini architect using the principles of quadratic equations!
Financial Planning
Have you ever wondered how much you need to save to buy that super cool bike in the future? Or how many lemonades do you need to sell at your stand to buy that new video game? Well, quadratic equations can help!
Also, these equations can help you determine how much money you need to save to reach your goal. Or, if you’re selling lemonades, they can help predict how many you’ll sell based on how much you’re charging.
So, next time you plan to save up for something or set up a lemonade stand, remember that quadratic equations are your secret tool to success!
Signal Processing
Ever tuned into your favorite radio station and danced to your favorite beats? Bet you didn’t know that quadratic equations play a part in that fun! They’re used in something called signal processing.
This might sound like a big, fancy word, but they help filter out the noise so the music sounds nice and clear. This way, you don’t hear any crackling or static when jamming to your favorite songs.
Quadratic equations ensure the tunes from your radio are smooth and perfect for your dance party!
Root Finding
Let’s talk about “Root Finding.” It may sound like we’re hunting for hidden treasure, but it’s a crucial part of solving our friend, the quadratic equation.
The “roots” are the special x values that balance the equation perfectly. Like solving a puzzle, finding these roots can crack open complex problems. For example, it helps scientists answer tricky questions in physics and lets engineers create cool stuff! But here’s the best part. It even helps make our video games more exciting.
Now, that’s pretty cool. So, the next time you play a game or see a fun experiment, remember these “roots” are like the hidden codes that make it all possible!
Computer Graphics
Get ready for this – quadratic equations are the secret ingredients that make your favorite video games and movies look super cool! Also, they help animators make things move and look real.
If a character needs to jump over a big chasm, a quadratic equation can help figure out how the character should move so it looks just like a real jump! And that’s not all. Quadratic equations are also used to determine how light and shadows should look.
This makes the game or movie feel like you’re really there. Also, it’s like being inside a game or movie! So, the next time you’re playing a video game or watching an animated movie, remember, there’s a bunch of quadratic equations making it all come to life!
How Can We Find the Roots of This Equation? 4x² – 5x – 12 = 0
Roots are the x-values that make an equation equal to zero. Finding them is important for solving quadratic equations.
There are a few main methods we can use. One way is to factor the equation. Also, when you factor a quadratic, it breaks into two binomials joined by a multiplication sign. Then, you set each factor equal to zero and solve.
Another method is to use the quadratic formula. Also, this formula relies on the coefficients (numbers above the x terms). Also, you plug the a, b, and c values into the formula to find the roots.
A third approach uses graphs. Also, by plotting the quadratic equation on a coordinate plane, the x-intercepts where it crosses the x-axis are the roots.
Technology like calculators can also help. Also, many have a “solve” or “root” feature to give the number values for the roots.
Being able to find the roots lets us fully solve any quadratic problem. Also, it’s an important part of understanding these flexible equations.
Analyzing the Roots of 4x² – 5x – 12 = 0
Here are some roots you need to know:
Root 1: What is the value of x for the positive root? 4x² – 5x – 12 = 0
Let’s say we have the quadratic equation: x^2 – 4x + 3 = 0.
To find the roots, we can use the quadratic formula. Also, plugging in the coefficients a=1, b=-4, and c=3 gives us:
- x = (-b ± √(b^2 – 4ac))/2a
- x = (4 ± √(16 – 12))/2
- x = (4 ± 4)/2
- x = 2, 3
This tells us that two roots, or values of x, make the equation equal to 0.
The positive root is where x equals the larger solution of 3. We write the positive root as x=3.
It’s important to identify signs of the roots correctly. Also, a positive root means the graph will cross the x-axis from bottom to top at that point.
Finding and analyzing the root values provides key details about the equation’s shape and behavior. Also, it helps us fully understand and solve the quadratic.
Root 2: What is the value of x for the negative root? 4x² – 5x – 12 = 0
Let’s analyze the roots of the same quadratic equation:
x^2 – 4x + 3 = 0
Using the quadratic formula, we found the two roots (or values of x that make the equation equal to 0) to be:
x = 2, 3
We already determined that the positive root is when x = 3.
To find the negative root, we look at the other solution:
Also, the negative root is when x = 2.
We can write the negative root as:
Root 2: x = 2
So, the value of x for the negative root of this quadratic equation is 2. Also, identifying the negative root correctly is important for understanding the full graph of the quadratic function.
Frequently Asked Questions About 4x² – 5x – 12 = 0
What’s a quadratic equation?
Well, it’s like a math puzzle with an x², an x, and a plain old number all mashed together. Also, it looks like this: ax² + bx + c = 0. Our example, 4x² – 5x – 12 = 0, is one of them!
What’s the deal with these terms you talked about?
These terms describe different parts of the equation. Also, the 4x² is the quadratic term, the -5x is the linear term, and the -12 is the constant term. Easy peasy!
Why do we call it a second-degree polynomial equation?
We call it that because the highest x power in the equation is 2. So, it’s like it graduated from the first degree to the second degree!
How do I solve this equation?
We’ll get to that part later. For now, remember that every quadratic equation can be solved, and it’s like a fun game to find the values of x!
Conclusion About 4x² – 5x – 12 = 0
By analyzing this quadratic equation term-by-term, we can understand its components and behavior. Also, the positive leading coefficient of 4x² – 5x – 12 = 0 indicates the parabolic shape will open upwards.
Using the quadratic formula allows us to solve for the two roots: x = 3 and x = -1. From these values, we can sketch the graph of the equation as a parabola that intersects the x-axis at (-1, 0) and (3, 0).
Factoring the equation into (4x – 3)(x + 4) confirms the roots. Also, decoding each part of the polynomial gives insight into its overall form and properties.
Examining the terms methodically helps build skills for solving more complex expressions and modeling real-world applications with higher-degree equations.
