Introduction:
A Quadratic Equation Is A Second-Order Polynomial Equation In A Single Variable X, With The General Form:
Ax2+Bx+C=0ax^2 + Bx + C = 0ax2+Bx+C=0
In This Context, We Will Solve The Quadratic Equation:
4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0
This Guide Will Take You Step-By-Step Through The Process Of Solving This Equation Using Different Methods.
Understanding The Equation:
The Given Equation Is:
4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0
Here, A=4a = 4a=4, B=−5b = -5b=−5, And C=−12c = -12c=−12.
Methods To Solve The Quadratic Equation:
Have you ever pondered how to unravel these enigmatic equations 4x^2 – 5x – 12 = 0 that appear in math class or infiltrate everyday scenarios? Well, fear not, for today, we embark on an exploration of the captivating domain of quadratic equations, with a special focus on one equation in particular: “4x ^ 2 – 5x – 12 = 0”.
But hold on, don’t let the numerical jumble intimidate you! Quadratic equations may sound daunting, but fret not – we’re here to navigate you through each step, making it as straightforward as a stroll in the park (or perhaps even simpler).
So, what’s the story behind “4x ^ 2 – 5x – 12 = 0”? Why does it hold significance, and why should you bother solving it? Join us as we unveil the mysteries of this equation, explore various methods to decipher its code, and uncover the pragmatic magic behind mastering quadratic equations. Prepare to enhance your mathematical prowess and unleash your inner problem-solving genius – let’s delve in!
What is this equation, “4x ^ 2 – 5x – 12 = 0”, and why is it crucial to solve?
Let’s delve into it! This equation, “4x ^ 2 – 5x – 12 = 0”, may seem like a random amalgamation of numbers and symbols, but it’s a distinctive type of mathematical puzzle known as a quadratic equation.
Now, what renders quadratic equations intriguing is their ubiquitous presence – from calculating the trajectory of a basketball in flight to predicting the path of a spacecraft in the cosmos. They serve as clandestine codes that unlock solutions to real-life conundrums!
But why is it imperative to solve this particular equation? Imagine yourself as a sleuth endeavoring to crack a case. Solving quadratic equations is akin to discovering clues – it aids us in comprehending mechanisms, making prognostications, and resolving real-world dilemmas.
Thus, whether you’re involved in constructing bridges, designing roller coasters, or simply seeking a deeper understanding of the world, mastering quadratic equations bestows upon you a superpower that facilitates unlocking the universe’s mysteries. Pretty remarkable, isn’t it? Let’s roll up our sleeves and embark on solving this equation to unleash our inner problem-solving superheroes!
Factoring Method:
Firstly, let’s attempt to factor the quadratic equation “4x ^ 2 – 5x – 12 = 0” into two binomial expressions. Here’s the process:
Step 1: Identify two numbers whose product equals the product of the coefficient of x^2 (which is 4) and the constant term (which is -12). Additionally, these numbers should sum up to the coefficient of x (which is -5).
Step 2: Upon finding the suitable pair of numbers, utilize them to rewrite the middle term (-5x) as the sum of two terms.
Step 3: Employ factor by grouping, pairing the terms, and determining the greatest common factor in each pair.
Step 4: Subsequently, apply the zero-product property by setting each factor equal to zero and solving for x.
Quadratic Formula:
If factoring appears intricate, fear not! We possess an alternative method – the quadratic formula. Here’s the procedure for employing it to solve “4x ^ 2 – 5x – 12 = 0”:
Step 1: Present the quadratic formula: x = (-b ± √(b^2 – 4ac)) / 2a, where a, b, and c denote the coefficients of the quadratic equation.
Step 2: Identify the values of a, b, and c in our equation “4x ^ 2 – 5x – 12 = 0”.
Step 3: Substitute these values into the quadratic formula and simplify.
Step 4: Determine the solutions for x by computing the two potential solutions using the ± symbol.
Completing the Square Method:
For those inclined towards algebraic manipulations, completing the square serves as another technique to crack “4x ^ 2 – 5x – 12 = 0”. Here’s the modus operandi:
Step 1: Reformulate the equation in the form (x – h)^2 = k.
Step 2: Ascertain the value of h by halving the coefficient of x and squaring the outcome.
Step 3: Add and subtract the value of h^2 within the parentheses to complete the square.
Step 4: Solve for x by extracting the square root of both sides and isolating x.
Graphical Method:
For visual learners, the graphical method might prove to be more favorable. Here’s how to tackle “4x^2 – 5x – 12 = 0” graphically:
Step 1: Plot the quadratic equation on a graph, with x on the horizontal axis and y on the vertical axis.
Step 2: Identify the x-intercepts, representing the solutions to the equation, by locating the points where the graph intersects the x-axis.
Step 3: Discern the values of x from the graph to ascertain the solutions.
By adhering to these systematic approaches, you’ll be equipped to solve “4x ^ 2 – 5x – 12 = 0” utilizing diverse methods, fostering a profound comprehension of quadratic equations. So, grasp your pencil, and let’s delve in!
Common Problems Faced in Solving “4x ^ 2 – 5x – 12 = 0:
While unraveling quadratic equations like “4x ^ 2 – 5x – 12 = 0” can be an intellectually gratifying endeavor, encountering obstacles is not uncommon. Here are some prevalent issues you might encounter, along with strategies to surmount them:
Difficulty Factoring:
Factoring quadratic equations necessitates identifying two numbers whose product yields the constant term and whose sum equals the coefficient of x. Occasionally, locating these numbers can be challenging, particularly if the equation features sizable coefficients or intricate terms.
Solution:
Invest ample time in exploring various factor pairs and consider employing techniques like trial and error or the AC method to streamline the factoring process. If factoring proves formidable, remember that alternative methods such as the quadratic formula or completing the square are at your disposal.
Complex Roots:
Quadratic equations may yield real or complex roots contingent upon the discriminant (the component within the square root in the quadratic formula). Confronting complex roots might appear daunting, particularly if you’re unaccustomed to imaginary numbers.
Solution:
Maintain composure! Complex roots hold equal validity as real roots and signify solutions to the equation within the complex number system. If grappling with complex numbers, allocate time to review fundamental concepts of imaginary numbers and their properties. Remember, practice is paramount!
Mistakes in Algebraic Manipulation:
Solving quadratic equations frequently entails numerous algebraic maneuvers, including simplifying expressions, amalgamating like terms, and isolating variables. Erroneous algebraic manipulations can engender incorrect solutions.
Solution:
Exercise diligence and meticulously review your work at each stage of the solving process. Pay heed to signs, correctly distribute terms and employ parentheses as needed to avert errors. If uncertain about a particular step, take a moment to revisit pertinent algebraic principles or solicit elucidation from an instructor or tutor.
Graphical Inaccuracies:
Though the graphical method may prove intuitively visual, inaccuracies in plotting points or interpreting values from the graph can yield erroneous solutions. Furthermore, quadratic equations with complex roots may lack easily discernible intercepts on the graph.
Solution:
Exercise caution when plotting points on the graph, resorting to graphing tools or software for precision if necessary. If the equation yields complex roots, contemplate employing alternative methods like factoring or the quadratic formula to pinpoint the solutions more accurately.
By cognizance of these common hurdles and deploying strategies to overcome them, you’ll be aptly equipped to confront quadratic equations like “4x ^ 2 – 5x – 12 = 0” with confidence and precision. Remember, persistence and practice are pivotal to mastering mathematical challenges!
Factoring:
Factoring Involves Expressing The Quadratic Equation As A Product Of Its Binomials. However, Not All Quadratic Equations Can Be Factored Easily.
Using The Quadratic Formula:
The Quadratic Formula Is The Most General Method For Solving Any Quadratic Equation. It Is Given By:
X=−B±B2−4ac2ax = \Frac{-B \Pm \Sqrt{B^2 – 4ac}}{2a}X=2a−B±B2−4ac
Completing The Square:
This Method Involves Transforming The Quadratic Equation Into A Perfect Square Trinomial.
Solving Using The Quadratic Formula:
The Quadratic Formula Is The Most Straightforward Approach For Solving Our Equation:
4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0
Step-By-Step, Let’s Solve It:
Identify Coefficients:
A=4,B=−5,C=−12a = 4, \Quad B = -5, \Quad C = -12a=4,B=−5,C=−12
Calculate The Discriminant:
Δ=B2−4ac\Delta = B^2 – 4acδ=B2−4ac Δ=(−5)2−4(4)(−12)\Delta = (-5)^2 – 4(4)(-12)Δ=(−5)2−4(4)(−12) Δ=25+192\Delta = 25 + 192Δ=25+192 Δ=217\Delta = 217Δ=217
Apply The Quadratic Formula:
X=−B±Δ2ax = \Frac{-B \Pm \Sqrt{\Delta}}{2a}X=2a−B±Δ X=5±2178x = \Frac{5 \Pm \Sqrt{217}}{8}X=85±217
So, The Solutions Are:
X=5+2178x = \Frac{5 + \Sqrt{217}}{8}X=85+217 X=5−2178x = \Frac{5 – \Sqrt{217}}{8}X=85−217
Verifying The Solutions:
Substituting The Solutions Back Into The Original Equation To Verify:
For X=5+2178x = \Frac{5 + \Sqrt{217}}{8}X=85+217:
4(5+2178)2−5(5+2178)−12=04 \Left(\Frac{5 + \Sqrt{217}}{8}\Right)^2 – 5 \Left(\Frac{5 + \Sqrt{217}}{8}\Right) – 12 = 04(85+217)2−5(85+217)−12=0
For X=5−2178x = \Frac{5 – \Sqrt{217}}{8}X=85−217:
4(5−2178)2−5(5−2178)−12=04 \Left(\Frac{5 – \Sqrt{217}}{8}\Right)^2 – 5 \Left(\Frac{5 – \Sqrt{217}}{8}\Right) – 12 = 04(85−217)2−5(85−217)−12=0
Both Values Satisfy The Equation, Confirming They Are Correct.
Solving By Completing The Square:
Completing The Square Is Another Method, But It Can Be More Complex For This Equation. For Completeness, Let’s Outline The Steps:
Rearrange The Equation:
4×2−5x=124x^2 – 5x = 124×2−5x=12
Divide By 4 To Simplify:
X2−54x=3x^2 – \Frac{5}{4}X = 3×2−45X=3
Add And Subtract (58)2\Left(\Frac{5}{8}\Right)^2(85)2 Inside The Equation:
X2−54x+(58)2=3+(58)2x^2 – \Frac{5}{4}X + \Left(\Frac{5}{8}\Right)^2 = 3 + \Left(\Frac{5}{8}\Right)^2×2−45X+(85)2=3+(85)2
Write As A Perfect Square:
(X−58)2=3+2564\Left(X – \Frac{5}{8}\Right)^2 = 3 + \Frac{25}{64}(X−85)2=3+6425 (X−58)2=192+2564\Left(X – \Frac{5}{8}\Right)^2 = \Frac{192 + 25}{64}(X−85)2=64192+25 (X−58)2=21764\Left(X – \Frac{5}{8}\Right)^2 = \Frac{217}{64}(X−85)2=64217
Take The Square Root Of Both Sides:
X−58=±21764x – \Frac{5}{8} = \Pm \Sqrt{\Frac{217}{64}}X−85=±64217 X−58=±2178x – \Frac{5}{8} = \Pm \Frac{\Sqrt{217}}{8}X−85=±8217
Solve For X:
X=58±2178x = \Frac{5}{8} \Pm \Frac{\Sqrt{217}}{8}X=85±8217 X=5+2178x = \Frac{5 + \Sqrt{217}}{8}X=85+217 X=5−2178x = \Frac{5 – \Sqrt{217}}{8}X=85−217
Conclusion:
The Quadratic Equation 4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0 Can Be Solved Using The Quadratic Formula Or By Completing The Square. The Solutions Are:
X=5+2178x = \Frac{5 + \Sqrt{217}}{8}X=85+217 X=5−2178x = \Frac{5 – \Sqrt{217}}{8}X=85−217
Both Methods Confirm The Same Solutions, Ensuring The Correctness Of Our Approach.