Quadratic Equations | Exercise 4.2

Question 4

Find two consecutive positive integers, sum of whose squares is 365.

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Solution
Understand the Question
  • Let the two consecutive positive integers be xx and x+1x + 1.
  • According to the problem, the sum of their squares is 365365, giving the equation x2+(x+1)2=365x^2 + (x + 1)^2 = 365.
  • We simplify this into a standard quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, solve for xx by factorisation, and select the positive integer value.

Step 1 · Formulate the Quadratic Equation

Let the two consecutive positive integers be xx and x+1x + 1.

Given

x2+(x+1)2=365x^2 + (x + 1)^2 = 365

Expanding (x+1)2(x + 1)^2

x2+(x2+2x+1)=3652x2+2x+1−365=02x2+2x−364=0\begin{aligned} x^2 + (x^2 + 2x + 1) &= 365 \\ 2x^2 + 2x + 1 - 365 &= 0 \\ 2x^2 + 2x - 364 &= 0 \end{aligned}

Dividing the entire equation by 22

x2+x−182=0x^2 + x - 182 = 0

Step 2 · Solve the Quadratic Equation

Factorising by splitting the middle term (14×−13=−18214 \times -13 = -182 and 14−13=114 - 13 = 1)

x2+14x−13x−182=0x(x+14)−13(x+14)=0(x−13)(x+14)=0\begin{aligned} x^2 + 14x - 13x - 182 &= 0 \\ x(x + 14) - 13(x + 14) &= 0 \\ (x - 13)(x + 14) &= 0 \end{aligned} x−13=0orx+14=0x - 13 = 0 \quad \text{or} \quad x + 14 = 0 x=13orx=−14x = 13 \quad \text{or} \quad x = -14

Since xx must be a positive integer, reject x=−14x = -14.

x=13x = 13

The consecutive integer is

x+1=13+1=14x + 1 = 13 + 1 = 14
Answer

1313 and 1414

Common Mistakes
  • Expansion Error: Writing (x+1)2=x2+1(x + 1)^2 = x^2 + 1 instead of x2+2x+1x^2 + 2x + 1, which omits the middle term 2x2x.
  • Ignoring the "Positive" Condition: Retaining the negative root x=−14x = -14. The question specifies positive integers, so x=−14x = -14 must be discarded.

More questions in Exercise 4.2

Q1

Find the roots of the following quadratic equations by factorisation:

(i) x2−3x−10=0x^2 - 3x - 10 = 0

(ii) 2x2+x−6=02x^2 + x - 6 = 0

(iii) 2x2+7x+52=0\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0

(iv) 2x2−x+18=02x^2 - x + \dfrac{1}{8} = 0

(v) 100x2−20x+1=0100x^2 - 20x + 1 = 0

Q2

Solve the problems given in Example 1.

Q3

Find two numbers whose sum is 27 and product is 182.

Q4

Find two consecutive positive integers, sum of whose squares is 365.

Q5

The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Q6

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹ 90, find the number of articles produced and the cost of each article.

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