Introduction:
Ax2+Bx+C=0ax^2 + Bx + C = 0ax2+Bx+C=0 Are Polynomial Equations Of Degree 2, Which Are Known As Quadratic Equations.
We Are Given The Quadratic Equation In This Instance:
04x^2 – 5x – 12 = 04×2 – 5x – 12 = 4×2 – 5x – 12 = 0
Step-By-Step Solution:
We Have Several Options For Solving The Quadratic Equation: Factoring, Completing The Square, And Applying The Quadratic Formula. Since The Quadratic Formula Is A Dependable Technique That May Be Used To Any Quadratic Problem, We Shall Employ It In This Instance.
Formula For Quadratic Equations:
Solving quadratic equations is a cornerstone of algebra and a foundational skill in mathematics. The equation “4x^2 – 5x – 12 = 0” presents a unique challenge and learning opportunity. This article delves deep into the techniques and strategies for solving this equation, ensuring that readers come away with a clear understanding and the ability to tackle similar problems.
Understanding Quadratic Equations:
Quadratic equations are polynomials of degree two, typically ax^2 + bx + c = 0, where a, b, and c are constants. The x represents the variable or unknown we are trying to solve. These equations are pivotal in various fields, from physics and engineering to finance and statistics.
A clear example is the quadratic equation in focus, 4x^2 – 5x – 12 = 0. Here, a = 4, b = -5, and c = -12. Understanding the structure of quadratic equations is crucial for solving them, as the values of a, b, and c directly influence the solving strategy and the nature of the solutions.
Factoring as a Strategy:
Factoring is one of the primary methods used to solve quadratic equations. It involves rewriting the quadratic in a product of two binomials. However, not all quadratics are easily factorable, especially when dealing with more significant coefficients or when the equation does not simplify neatly. Regarding 4x^2 – 5x – 12 = 0, factoring can be challenging due to the coefficients involved. Nevertheless, one can uncover the factors that solve the equation with some practice and technique.
The Process of Factoring:
Factoring aims to find two numbers that multiply to give the product of a and c and add to provide b. For our equation, we look for two numbers that multiply to (4)(-12) = -48 and add to -5. This step requires trial and error but is facilitated by a strong understanding of the number of properties and factors.
Once the appropriate numbers are found, the equation is rewritten in a factored form, which can then be set to zero to solve for the values of x. This method is highly effective for easily factorable equations, providing a quick and straightforward solution.
The Quadratic Formula: A Universal Solver:
When factoring is too cumbersome or impossible, the quadratic formula offers a fail-safe solution. This formula, x = [-b ± √(b^2 – 4ac)] / 2a, works for any quadratic equation. It directly utilizes the coefficients a, b, and c to find the solutions of x.
Applying the Quadratic Formula to 4x^2 – 5x – 12 = 0:
To solve 4x^2 – 5x – 12 = 0 using the quadratic formula, we substitute the values of a, b, and c into the formula. This process demystifies the equation, turning it into a straightforward calculation. The beauty of the quadratic formula lies in its universality and reliability, ensuring that even the most complex equations can be solved.
Completing the Square: An Alternate Route:
Completing the square is another technique for solving quadratic equations. It involves transforming the equation into a perfect square trinomial, from which the value of x can be easily derived. This method is beneficial for understanding the quadratic formula’s derivation and solving equations when other methods are more complex.
How to Complete the Square:
Completing the square requires rearranging and manipulating the equation so that the left side becomes a perfect square. The equation 4x^2 – 5x – 12 = 0 involves:
- Dividing by the leading coefficient (if not 1).
- Moving the constant term to the other side.
- Adding a specific value to both sides to create a perfect square trinomial.
This method finds the solution and offers insight into the equation’s structure.
Graphical Solutions: Visualizing the Equation:
Graphing is a powerful tool for visualizing and solving quadratic equations. Plotting the equation on a coordinate plane allows one to see the parabola it forms and identify its roots—the points at which it intersects the x-axis.
Solving 4x^2 – 5x – 12 = 0 Graphically:
To solve our equation graphically, one would plot y = 4x^2 – 5x – 12 and look for the x-values where y equals zero. This approach provides a visual understanding of the solutions and the behavior of quadratic functions, complementing the algebraic techniques.
Here Is The Quadratic Formula:
\Frac{-B \Pm \Sqrt{B^2 – 4ac}} = X=−B±B2−4ac2ax{2a}X=2a−B±B2−4ac}, Where The Coefficients Of The Quadratic Equation Ax2+Bx+C=0ax^2 + Bx + C = 0ax2+Bx+C=0 Are Aaa, Bbb, And Ccc…
The Coefficients For Our Equation 4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0 Are As Follows:
• A=4a = 4a=4
• B=-5b = -5b = −5
• C=-12c = -12c = -12
Step 1: Determine The Discriminant:
The Discriminant Δ\Deltaδ Is Computed As Follows And Is A Component Of The Quadratic Formula Under The Square Root:
Δ=B2−4acdelta Is Equal To B^2 – 4ac.Δ=B2−4ac
Changing The Values Of Bbb, Ccc, And Aaa:
Δ=(−5)2−4(4)(−12)\(-5)^2 – 4(4)(-12) Is Delta.Δ=(−5)2−4(4)(−12) Δ=25+192\Delta Is Equal To 25 Plus 192.Δ=25+192 Δ=217\Delta = 217Δ = 217
We Get Two Different Real Solutions Since The Discriminant Is Positive (Δ>0\Delta > 0Δ>0).
Step 2: Utilize The Formula For Quadratics:
Enter Aaa, Bbb, And Δ\Deltaδ Values Into The Quadratic Formula Now:
X=Frac{-(-5) \Pm \Sqrt{217}} = −(−5)±2172(4){2(4)}X=5±2178x = \Frac{5 \Pm \Sqrt{217}}{8}X=85±217; X=2(4)−(−5)±217
Step 3: Make The Solutions Simpler
The Answers Are As Follows:
X2=5−2178x_2 = \Frac{5 – \Sqrt{217}}{8}X2=85−217 X1=5+2178x_1 = \Frac{5 + \Sqrt{217}}{8}X1=85+217
Checking The Solutions:
We Can Reintroduce Our Solutions Into The Original Equation And Check That They Fulfill It To Make Sure Our Solutions Are Accurate.
Confirmation Regarding X1x_1x1:
The Equation X1=5+2178x_1 = \Frac{5 + \Sqrt{217}}{8}X1=85+217
To Solve 4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0, Substitute X1x_1x1 Into 4(5+2178)2−5(5+2178)5\Left(\Frac{5 + \Sqrt{217}}{8}\Right) – −124\Left(\Frac{5 + \Sqrt{217}}{8}\Right)^2 – 124 (85 + 217)2−5(85+217)-12
This Reduces To Zero, Indicating That X1x_1x1 Is A Solution.
Confirmation Regarding X2x_2x2:
\Frac{5 – \Sqrt{217}}{8}X2=85−217; X2=5−2178x_2
In 4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0, Substitute X2x_2x2: 4(5−2178)2−5(5−2178)−124\Left(\Frac{5 – \Sqrt{217}}{8}\Right)^2 – 5\Left(\Frac{5 – \Sqrt{217}}{8}\Right)^2 – 5\Left(\Frac{5 – \Sqrt{217}}{8}\Right) – 124(85- 217)2−5(85- 217)−12
This Likewise Reduces To 0, Indicating That X2x_2x2 Is A Solution.
Summary:
4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0 Has The Following Solutions: X1=5+2178x_1 = \Frac{5 + \Sqrt{217}}{8}X1=85+217 X2=5−2178x_2 = \Frac{5 – \Sqrt{217}}{8}X2=85−217
The Quadratic Formula Was Used To Find These Answers, Which Were Then Confirmed By Substituting Them Back Into The Original Problem.