STATEMENT - 1: $ (x-2)(x+1)=(x-1)(x+3) $ is a quadratic equation.
STATEMENT - 2: If $ p(x) $ is a quadratic polynomial, then $ p(x)=0 $ is called a quadratic equation.
A. Statement - 1 is True, Statement - 2 is True. Statement - 2 is a correct explanation for Statement - 1.
B. Statement - 1 is True, Statement - 2 is True. Statement - 2 is NOT a correct explanation for Statement
- 1.
C. Statement - 1 is True, Statement - 2 is False.
D. Statement - 1 is False, Statement - 2 is True.
Answer
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Hint:For a quadratic expression (polynomial), the degree of the expression must be 2.
What is the difference between "a polynomial " and "an equation "?
Simplify the expressions into monomials to find their degree.
The case of checking whether one of the statements is an explanation for the other or not, arises only when both the statements are true.
Complete step by step solution:
Let’s look at the statements one by one.
STATEMENT - 1: $ (x-2)(x+1)=(x-1)(x+3) $ is a quadratic equation. FALSE.
If we multiply the terms on both sides of the equality, we get:
⇒ $ {{x}^{2}}-x-2={{x}^{2}}+2x-3 $
On subtracting RHS from both the sides, we get:
⇒ $ -3x+1=0 $
Which is a linear equation because its degree is 1. Hence, the statement is false.
STATEMENT - 2: If $ p(x) $ is a quadratic polynomial, then $ p(x)=0 $ is called a quadratic equation.
TRUE.
When a polynomial is equated with another polynomial or a constant, it becomes an equation (equality of two expressions).
The correct answer is D. Statement - 1 is False, Statement - 2 is True.
Note:The graph of a univariate (single variable) quadratic polynomial is a parabola. In general, the graph of a bivariate quadratic polynomial is a conic section (parabola, circle, ellipse, hyperbola or a pair of intersecting lines).
The degree of a polynomial is the highest of the degrees of the polynomial individual terms with non- zero coefficients. In order to find the degree of an expression/equation, the variables must be freed of radicals and rational forms.
The degree of terms involving a product of variables is NOT 1.
What is the difference between "a polynomial " and "an equation "?
Simplify the expressions into monomials to find their degree.
The case of checking whether one of the statements is an explanation for the other or not, arises only when both the statements are true.
Complete step by step solution:
Let’s look at the statements one by one.
STATEMENT - 1: $ (x-2)(x+1)=(x-1)(x+3) $ is a quadratic equation. FALSE.
If we multiply the terms on both sides of the equality, we get:
⇒ $ {{x}^{2}}-x-2={{x}^{2}}+2x-3 $
On subtracting RHS from both the sides, we get:
⇒ $ -3x+1=0 $
Which is a linear equation because its degree is 1. Hence, the statement is false.
STATEMENT - 2: If $ p(x) $ is a quadratic polynomial, then $ p(x)=0 $ is called a quadratic equation.
TRUE.
When a polynomial is equated with another polynomial or a constant, it becomes an equation (equality of two expressions).
The correct answer is D. Statement - 1 is False, Statement - 2 is True.
Note:The graph of a univariate (single variable) quadratic polynomial is a parabola. In general, the graph of a bivariate quadratic polynomial is a conic section (parabola, circle, ellipse, hyperbola or a pair of intersecting lines).
The degree of a polynomial is the highest of the degrees of the polynomial individual terms with non- zero coefficients. In order to find the degree of an expression/equation, the variables must be freed of radicals and rational forms.
The degree of terms involving a product of variables is NOT 1.
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