Many word problems Involving unknown quantities can be translated for solving quadratic equations
Methods of solving quadratic equations are discussed here in the following steps.
Step I: Denote the unknown quantities by x, y etc.
Step II: use the conditions of the problem to establish in unknown quantities.
Step III: Use the equations to establish one quadratic equation in one unknown.
Step IV: Solve this equation to obtain the value of the unknown in the set to which it belongs.
Now we will learn how to frame the equations from word problem:
1. The product of two consecutive integers is 132. Frame an equation for the statement. What is the degree of the equation?
Solution:
Method I: Using only one unknown
Let the two consecutive integers be x and x + 1
Form the equation, the product of x and x + 1 is 132.
Therefore, x(x + 1) = 132
⟹ x\(^{2}\) + x  132 = 0, which is quadratic in x.
This is the equation of the statement, x denoting the smaller integer.
Method II: Using more than one unknown
Let the consecutive integers be x and y, x being the smaller integer.
As consecutive integers differ by 1, y  x = 1 ........................................... (i)
Again, from the question, the product of x and y is 132.
So, xy = 132 ........................................... (ii)
From (i), y = 1 + x.
Putting y = 1 + x in (ii),
x(1 + x) = 132
⟹ x\(^{2}\) + x  132 = 0, which is quadratic in x.
Solving the quadratic equation, we get the value of x. Then the value of y can be determined by substituting the value of x in y = 1 + x.
2. The length of a rectangle is greater than its breadth by 3m. If its area be 10 sq. m, find the perimeter.
Solution:
Suppose, the breadth of the rectangle = x m.
Therefore, length of the rectangle = (x + 3) m.
So, area = (x + 3)x sq. m
Hence, by the condition of the problem
(x + 3)x = 10
⟹ x\(^{2}\) + 3x  10 = 0
⟹ (x + 5)(x  2) = 0
So, x = 5,2
But x =  5 is not acceptable, since breadth cannot be negative.
Therefore x = 2
Hence, breadth = 2 m
and length = 5 m
Therefore, Perimeter = 2(2 + 5) m = 14 m.
x = 5 does not satisfy the conditions of the problem length or breadth can never be negative. Such a root is called an extraneous root. In solving a problem, each root of the quadratic equation is to be verified whether it satisfies the conditions of the given problem. An extraneous root is to be rejected.
Algebra/Linear Algebra
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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