Introduction:
Algebraic Quadratic Equations Are Basic, And Solving Them Is Essential For Many Applications In Science And Mathematics. This Tutorial Will Walk You Through The Process Of Solving The Quadratic Equation 4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0 Step-By-Step. Various Techniques Will Be Covered, Along With Thorough Explanations And Examples To Help With Each Stage.
Understanding The Quadratic Equation:
What Is A Quadratic Equation?
A Second-Degree Polynomial Equation With The Formula Ax2+Bx+C=0ax^2 + Bx + C = 0ax2+Bx+C=0 Is Called A Quadratic Equation.
Where Xxx Is The Variable And Aaa, Bbb, And Ccc Are The Constants.
Particulars Of Our Formula:
In The Formula 04x^2 – 5x – 12 = 04×2 – 5x – 12 = 4×2 – 5x – 12 = 0:
- A=4a = 4a=4
- B=-5b = -5b = −5
- C=-12c = -12c = -12
Approaches To Quadratic Equation Solving:
Factoring:
The Factoring Process:
4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0 Is The Supplied Equation.
Look For Two Numbers:
Find Two Numbers That Add Up To Bbb (I.E., −5-5−5) And Multiply To A⋅Ca \Cdot Ca⋅C (I.E., 4×−12=−484 \Times -12 = -484×−12=−48).
The Reasons These Numbers Are −8–8−8 And 666 Are:
-48−8×6=−48 −8+6=−5-8 + 6 = -5−8+6=−5
Modify The Intermediate Term:
Put −5x-5x−5x In The Form −8x+3x-8x + 3x−8x+3x.
4x^2 – 8x + 3x – 12 = 04×2 – 8x + 3x – 12 = 0
Factor Based On Grouping:
Sort The Terms:
(4×2−8x) + (3x−12) = 04x^2 – 8x) + (3x – 12) = 0 (4x^2 – 8x) + (3x – 12) = 0
Determine The Commonalities:
4x(-2)+3(-X)=04x(-2) + 3(-X) = 04x(-2)+3(-X)=0
The Fifth Common Binomial Factor:
Put The Terms Together To Get:
(4x+3)(X-4) = 0(4 X + 3)(X – 4) = 0(4 X + 3)(X – 4) = 0
Find The Solution For Xxx:
Put A Zero In Each Factor And Solve:
\Text{Or} \Quad X – 4 = 04x+3=0orx−4=0 \Quad 4x+3=0orx−4=04x + 3 = 0 X = 44x = −3orx=4 \Quad \Text{Or} \Quad X = 4x = −3orx=44 -\Frac{3}{4} \Quad \Text{Or} \Quad X = 4x=−43orx=4 X=−34orx=4
X=−34x = -\Frac{3}{4}X=−43 And X=4x = 4x=4 Are The Solutions.
2.2 Application Of The Quadratic Formula:
The Quadratic Formula’s Steps:
Here Is The Quadratic Formula:
X = -B \Pm \Sqrt{B^2 – 4ac}}{2a}X=2a−B±B2−4ac
Recognize Ccc, Bbb, And Aaa:
In The Case Of 4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0a = 4, \Quad B = -5, \Quad C = -12; A = 4, B = −5, C = −12
Determine Who Is Discriminant:
Δ=B2−4acdelta Is Equal To B^2 – 4ac.Δ=B2−4ac Δ=(−5)2−4⋅4⋅(−12)\(-5)^2 – 4 \Cdot 4 \Cdot (-12) = Deltaδ=(−5)2−4⋅4⋅(−12) Δ=25+192\Delta Is Equal To 25 Plus 192.Δ=25+192 Δ=217\Delta = 217Δ = 217
Calculate The Answers:
X = -(-5) \Pm \Sqrt{217}}{2 \Cdot 4} = −(−5)±2172⋅4xx=5±2178x = \Frac{5 \Pm \Sqrt{217}}{8} X=2⋅4−(−5)±217x Equals 85±217
The Precise Answers Are As Follows:
X = 5 + \Sqrt{217}}{8} \Quad \Text{And} \Quad X = \Frac{5 – \Sqrt{217}}{8} For X = 5+2178 And X = 5−2178.X = 85 + 217 And X = 85 – 217
Finishing Up The Square:
Procedure For Finishing The Square:
Rephrase The Formula:
04x^2 – 5x – 12 = 04×2 – 5x – 12 = 4×2 – 5x – 12 = 0
Transfer The Constant Word To The Opposite Side:
12 = 124×2 – 5x = 4×2 – 5x = 124×2 – 5x
Split According To Aaa:
\Frac{5}{4}X = 3×2−45x=3 – X2−54x=3x^2
Finish The Square:
(B2)2\Left(\Frac{B}{2}\Right)^2(2b)2 Is Added And Subtracted:
X2−54x+(58)2=3 + (58)2x^2 – \Left(\Frac{5}{8}\Right) + \Frac{5}{4}X3 + \Left(\Frac{5}{8}\Right) = ^23+(85)±2×2−45x+(85)22 (X-58)2 = 16964\Right\Frac{169}{64}(X−85)^2 = (X – \Frac{5}{8}\Right)^2
XXX Solution:
X−58=±16964x – \Frac{5}{8} = \Pm \Sqrt{\Frac{169}{64}}X−85=±64169 X−58=±138x – \Frac{5}{8} = \Pm \Frac{13}{8}X−85=±813 X=58±138x = \Frac{5}{8} \Pm \Frac{13}{8}X=85±813
The Answers Are As Follows:
X = \Frac{18}{8} = \Frac{9}{4} \Quad \Text{And} \Quad X = -\Frac{8}{8} = -1x=818=49andx=−88=−1x = \Frac{18}{8}
Confirmation Of Solutions:
Factoring Technique:
In Place Of X=−34x = -\Frac{3}{4}X=−43, Use X=4x = 4x=4. To Confirm, Go Back Into The Original Formula.
Method Of The Quadratic Formula:
To Determine Whether The Resultant Values Meet 4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0, Substitute Them Into The Original Equation.
Finishing The Square Technique:
To Confirm Accuracy, Replace X=94x = \Frac{9}{4}X=49 And X=−1x = -1x=−1 Back Into The Equation.
Final Thought:
The Ability To Solve Quadratic Equations Is A Fundamental Algebraic Skill. Three Distinct Approaches Have Been Covered In This Guide: Factoring, The Quadratic Formula, And Completing The Square. There Are Several Approaches To Solving The Equation 4×2−5x−12=04x^2 – 5x – 12 = 04×2−5x−12=0, And Being Aware Of Them All Aids In Selecting The Most Effective One For The Given Situation.
