Which of the following is NOT a quadratic equation?
To determine which of the given equations is NOT a quadratic equation, we need to simplify each equation and check if it can be written in the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\), where \(a, b, c\) are real numbers and \(a \neq 0\).
Let's analyze each option:
Analyzing Option 1 to Identify Quadratic Equations
The given equation is \((x + 1)^2 = 2(x - 3)\).
Let's expand and simplify this equation:
- Expand the left side using the identity \((a+b)^2 = a^2 + 2ab + b^2\):
\((x + 1)^2 = x^2 + 2(x)(1) + 1^2 = x^2 + 2x + 1\) - Expand the right side:
\(2(x - 3) = 2x - 6\) - Now, equate the expanded sides:
\(x^2 + 2x + 1 = 2x - 6\) - Move all terms to one side to set the equation to zero:
\(x^2 + 2x + 1 - 2x + 6 = 0\) - Combine like terms:
\(x^2 + (2x - 2x) + (1 + 6) = 0\) - Simplify:
\(x^2 + 0x + 7 = 0\) - This simplifies to \(x^2 + 7 = 0\).
This equation is in the form \(ax^2 + bx + c = 0\), with \(a=1\), \(b=0\), and \(c=7\). Since \(a = 1 \neq 0\), this is a quadratic equation.
Analyzing Option 2 to Identify Quadratic Equations
The given equation is \((x + 2)^2 = 2x(x + 1)\).
Let's expand and simplify this equation:
- Expand the left side using the identity \((a+b)^2 = a^2 + 2ab + b^2\):
\((x + 2)^2 = x^2 + 2(x)(2) + 2^2 = x^2 + 4x + 4\) - Expand the right side:
\(2x(x + 1) = 2x \cdot x + 2x \cdot 1 = 2x^2 + 2x\) - Now, equate the expanded sides:
\(x^2 + 4x + 4 = 2x^2 + 2x\) - Move all terms to one side:
\(x^2 + 4x + 4 - 2x^2 - 2x = 0\) - Combine like terms:
\((x^2 - 2x^2) + (4x - 2x) + 4 = 0\) - Simplify:
\(-x^2 + 2x + 4 = 0\)
This equation is in the form \(ax^2 + bx + c = 0\), with \(a=-1\), \(b=2\), and \(c=4\). Since \(a = -1 \neq 0\), this is a quadratic equation.
Analyzing Option 3 to Identify Quadratic Equations
The given equation is \(x^2 + 3x + 1 = (x - 2)^2\).
Let's expand and simplify this equation:
- The left side is already simplified: \(x^2 + 3x + 1\).
- Expand the right side using the identity \((a-b)^2 = a^2 - 2ab + b^2\):
\((x - 2)^2 = x^2 - 2(x)(2) + 2^2 = x^2 - 4x + 4\) - Now, equate the expanded sides:
\(x^2 + 3x + 1 = x^2 - 4x + 4\) - Move all terms to one side:
\(x^2 + 3x + 1 - x^2 + 4x - 4 = 0\) - Combine like terms:
\((x^2 - x^2) + (3x + 4x) + (1 - 4) = 0\) - Simplify:
\(0x^2 + 7x - 3 = 0\) - This simplifies to \(7x - 3 = 0\).
This equation is in the form \(ax^2 + bx + c = 0\), but here \(a=0\), \(b=7\), and \(c=-3\). Since the coefficient of the \(x^2\) term is \(0\), this equation is a linear equation, NOT a quadratic equation.
Analyzing Option 4 to Identify Quadratic Equations
The given equation is \(m(2m + 3) = m^2 + 1\). Note that the variable here is \(m\).
Let's expand and simplify this equation:
- Expand the left side:
\(m(2m + 3) = m \cdot 2m + m \cdot 3 = 2m^2 + 3m\) - The right side is already simplified: \(m^2 + 1\).
- Now, equate the expanded sides:
\(2m^2 + 3m = m^2 + 1\) - Move all terms to one side:
\(2m^2 + 3m - m^2 - 1 = 0\) - Combine like terms:
\((2m^2 - m^2) + 3m - 1 = 0\) - Simplify:
\(m^2 + 3m - 1 = 0\)
This equation is in the form \(am^2 + bm + c = 0\) (standard quadratic form with variable \(m\)), with \(a=1\), \(b=3\), and \(c=-1\). Since \(a = 1 \neq 0\), this is a quadratic equation.
Conclusion on Quadratic Equations
Based on the analysis, Option 3 simplifies to a linear equation (\(7x - 3 = 0\)) because the \(x^2\) terms cancel out, resulting in the coefficient of \(x^2\) being zero. The other options all simplify to equations of the form \(ax^2 + bx + c = 0\) where \(a \neq 0\), making them quadratic equations.
| Option | Original Equation | Simplified Form | Value of \(a\) (coefficient of \(x^2\) or variable squared) | Is it Quadratic? |
|---|---|---|---|---|
| 1 | \((x + 1)^2 = 2(x - 3)\) | \(x^2 + 7 = 0\) | \(1\) | Yes |
| 2 | \((x + 2)^2 = 2x(x + 1)\) | \(-x^2 + 2x + 4 = 0\) | \(-1\) | Yes |
| 3 | \(x^2 + 3x +1 = (x - 2)^2\) | \(7x - 3 = 0\) | \(0\) | No |
| 4 | \(m(2m + 3) = m^2 + 1\) | \(m^2 + 3m - 1 = 0\) | \(1\) | Yes |
Therefore, the equation that is NOT a quadratic equation is \(x^2 + 3x + 1 = (x - 2)^2\).
Revision Table: Key Concepts for Quadratic Equations
| Concept | Description | Standard Form |
|---|---|---|
| Quadratic Equation | A polynomial equation of the second degree. | \(ax^2 + bx + c = 0, \text{ where } a \neq 0\) |
| Linear Equation | A polynomial equation of the first degree. | \(bx + c = 0, \text{ where } b \neq 0\) |
| Coefficient 'a' | The number multiplying the squared term (e.g., \(x^2\)). For a quadratic equation, this must be non-zero. | In \(ax^2 + bx + c = 0\), 'a' is the coefficient of \(x^2\). |
Additional Information: Understanding Equation Types
Equations are classified based on the highest power of the variable they contain after simplification. This highest power is called the degree of the polynomial.
- Degree 1: Linear Equation
Highest power is 1 (e.g., \(x^1\) or just \(x\)). Standard form: \(ax + b = 0\) where \(a \neq 0\). Example: \(2x + 5 = 0\).
- Degree 2: Quadratic Equation
Highest power is 2 (e.g., \(x^2\)). Standard form: \(ax^2 + bx + c = 0\) where \(a \neq 0\). Example: \(3x^2 - 4x + 1 = 0\).
- Degree 3: Cubic Equation
Highest power is 3 (e.g., \(x^3\)). Standard form: \(ax^3 + bx^2 + cx + d = 0\) where \(a \neq 0\). Example: \(x^3 - 2x^2 + x - 5 = 0\).
When simplifying an equation, if the terms with the highest power cancel out, the degree of the equation becomes lower. In the case of Option 3, the \(x^2\) terms cancelled out, reducing it from a potential quadratic (degree 2) to a linear equation (degree 1).
Important Questions from Quadratic Equation
If the equations x 2+ ax + b = 0 and x 2+ bx + a = 0 have a common root, then find the value of a + b (where a is not equal to b)
If x 2+ 1 = 2x, then find x – \((\frac{1}{x})\)
- Roots of the following equation are 6x 2+ 4x - 2 = 0
Solve : (x + 2y) (2x – y)
A. 2x 2+ 5xy – 2y 2
B. 2x 2+ 3xy – 2y 2
C. x 2+ 4xy + y 2
D. x 2+ 4xy – y 2
Find the factors of (x 2– x – 132)?
