We will learn how to solve Word Problems on quadratic equations by factoring.
1. The product of two numbers is 12. If their sum added to the sum of their squares is 32, find the numbers.
Solution:
Let the numbers be x and y.
As their product is 12, we get xy = 12 ..................... (i)
According to the question, x + y + x\(^{2}\) + y\(^{2}\) = 32 ..................... (ii)
From (i), y = \(\frac{12}{x}\)
Putting y = \(\frac{12}{x}\) in (ii), we get
x + \(\frac{12}{x}\) + x\(^{2}\) + (\(\frac{12}{x}\))\(^{2}\) = 32
⟹ (x + \(\frac{12}{x}\)) + (x + \(\frac{12}{x}\))\(^{2}\) - 2 x ∙ \(\frac{12}{x}\) = 32
⟹ (x + \(\frac{12}{x}\))\(^{2}\) + (x + \(\frac{12}{x}\)) - 56 = 0
Putting x + \(\frac{12}{x}\) = t,
t\(^{2}\) + t - 56 = 0
⟹ t\(^{2}\) + 8t – 7t – 56 = 0
⟹ t(t + 8) - 7(t + 8) = 0
⟹ (t + 8)(t - 7) = 0
⟹ t + 8 = 0 or, t – 7 = 0
⟹ t = -8 or, t = 7
When t = -8,
x + \(\frac{12}{x}\) = t = -8
⟹ x\(^{2}\) + 8x + 12 = 0
⟹ x\(^{2}\) + 6x + 2x + 12 = 0
⟹ x(x + 6) + 2(x + 6) = 0
⟹ (x + 6)(x + 2) = 0
⟹ x + 6 = 0 or, x + 2 = 0
⟹ x = -6 or, x = -2
When t = 7
x + \(\frac{12}{x}\) = t = 7
⟹ x\(^{2}\) - 7x + 12 = 0
⟹ x\(^{2}\) - 4x - 3x + 12 = 0
⟹ x(x – 4) - 3(x – 4) = 0
⟹ (x - 4)(x - 3) = 0
⟹ x - 4 = 0 or, x - 3 = 0
⟹ x = 4 or 3
Thus, x = -6, -2, 4, 3
Then, the other number y = \(\frac{12}{x}\) = \(\frac{12}{-6}\), \(\frac{12}{-2}\), \(\frac{12}{4}\), \(\frac{12}{3}\) = -2, -6, 3, 4.
Thus, the two numbers x, y are -6, -2, or -2, -6, or 4, 3 or 3, 4.
Therefore, the required two numbers are -6, -2 or 4, 3.
2. An association has a fund of $195. In addition that, each member of the association contributes the number of dollars equal to the number of members. The total money is divided equally among the members. If each of the members gets $ 28, find the number of members in the association.
Solution:
Let the number of members be x.
Total contributions from them = $ x\(^{2}\) and the association has a fund of $ 195.
According to the problem,
x\(^{2}\) + 195 = 28x
⟹ x\(^{2}\) - 28x + 195 = 0
⟹ x\(^{2}\) - 15x - 13x + 195 = 0
⟹ x(x - 15) - 13(x - 15) = 0
⟹ (x - 15)(x - 13) = 0
Therefore, x = 15 or 13
There are 15 or 13 members in the association.
Note: Two answers are acceptable in this case.
Quadratic Equation
Introduction to Quadratic Equation
Formation of Quadratic Equation in One Variable
General Properties of Quadratic Equation
Methods of Solving Quadratic Equations
Examine the Roots of a Quadratic Equation
Problems on Quadratic Equations
Quadratic Equations by Factoring
Word Problems Using Quadratic Formula
Examples on Quadratic Equations
Word Problems on Quadratic Equations by Factoring
Worksheet on Formation of Quadratic Equation in One Variable
Worksheet on Quadratic Formula
Worksheet on Nature of the Roots of a Quadratic Equation
Worksheet on Word Problems on Quadratic Equations by Factoring
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