Theory of Intelligence
11TH•Mathematics•Theory Of Quadratic Functions•Exercise 3.2

3.2 Inverse of Quadratic Function

Practice set 3.2 (equations reducible to quadratic; rational/radical equations and applications).

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Q1

Question 1

Solve the following equations:

Select Part:Part (i) of 8
(i)

13x+4x6=1,  x≠0\dfrac{1}{3x}+\dfrac{4x}{6}=1,\; x\ne 0

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Step-by-Step Solution

Question 1 • Part (i)

verifiedVerified Answer

Solve the equation: 13x+4x6=1(x≠0)\frac{1}{3x} + \frac{4x}{6} = 1 \quad (x \ne 0)

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Algebraic Lab•Interactive

Equation Balance Scale & Solution Verifier

visibilityArrange Display View:
Slide Test Value (x):
x = 1.00
-2Domain: x ∈ ℝ \ {0} (division by zero at x = 0)3
Test Candidate:
balancePhysical Balance Scale (LHS vs RHS)✓ EQUATION BALANCED
LHSRHS
Left Hand Side (LHS)
1.000
Right Hand Side (RHS)
1.000
Balance Status
✓ EQUAL ROOT INTERSECTION
Exact Algebraic Root Verified at x = 1.00:

Both sides of the equation balance at exactly 1.000. The beam rests perfectly horizontal, confirming this is a genuine root.

show_chartDual-Curve Geometric Intersections
y=13x+2x3y = \frac{1}{3x} + \frac{2x}{3}y=1y = 1
02x = 1/2x = 1

Step-by-Step Solution

Step 1: Simplify and clear fractions

Multiply the entire equation by the common denominator 6x6x: 6x(13x)+6x(4x6)=6x(1)6x \left(\frac{1}{3x}\right) + 6x \left(\frac{4x}{6}\right) = 6x(1)

2+4x2=6x2 + 4x^2 = 6x

Step 2: Write in standard quadratic form

4x2−6x+2=04x^2 - 6x + 2 = 0

Divide by 22: 2x2−3x+1=02x^2 - 3x + 1 = 0

Step 3: Factorize

2x2−2x−x+1=02x^2 - 2x - x + 1 = 0 2x(x−1)−1(x−1)=02x(x - 1) - 1(x - 1) = 0 (2x−1)(x−1)=0(2x - 1)(x - 1) = 0

2x−1=0  ⟹  x=122x - 1 = 0 \implies x = \frac{1}{2} x−1=0  ⟹  x=1x - 1 = 0 \implies x = 1

Step 4: Check solutions

  • For x=12x = \frac{1}{2}: 13(1/2)+4(1/2)6=23+13=1\dfrac{1}{3(1/2)} + \dfrac{4(1/2)}{6} = \dfrac{2}{3} + \dfrac{1}{3} = 1 ✓\checkmark
  • For x=1x = 1: 13(1)+4(1)6=13+23=1\dfrac{1}{3(1)} + \dfrac{4(1)}{6} = \dfrac{1}{3} + \dfrac{2}{3} = 1 ✓\checkmark

Solution Set: {12,  1}\boxed{\text{Solution Set: } \left\{\frac{1}{2}, \; 1\right\}}

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