Problem 107 - Algebra for College Students

Question

Which of the following equations is not a quadratic equation? Explain your answer. a) \(\pi x^{2}-\sqrt{5} x-1=0\) b) \(3 x^{2}-1=0\) c) \(4 x+5=0\) d) \(0.009 x^{2}=0\)

Step-by-Step Solution

Answer

Option c) (4x + 5 = 0) is not a quadratic equation.

Step 1 - Identify the Standard Form of a Quadratic Equation

A quadratic equation is generally in the form \[ ax^{2} + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \).

Step 2 - Analyze Option a

For option a), \( \pi x^{2} - \sqrt{5} x - 1 = 0 \), the equation is in the standard quadratic form where \(a = \pi \), \( b = -\sqrt{5} \), and \( c = -1 \). Therefore, it is a quadratic equation.

Step 3 - Analyze Option b

For option b), \( 3 x^{2} - 1 = 0 \), the equation is in the standard quadratic form where \( a = 3 \) and \( c = -1 \). It lacks the \( bx \) term, but it is still a quadratic equation.

Step 4 - Analyze Option c

For option c), \( 4 x + 5 = 0 \), identify the powers of \( x \). The equation has no \( x^{2} \) term, making it a linear equation, not a quadratic one.

Step 5 - Analyze Option d

For option d), \( 0.009 x^{2} = 0 \), the equation is in the standard quadratic form where \( a = 0.009 \). Despite having no \(bx \) term, it is still a quadratic equation due to the \( x^{2} \) term.

Step 6 - Conclusion

Based on the analysis, only option c) does not feature a \( x^{2} \) term and is therefore not a quadratic equation.

Key Concepts

Standard Form of Quadratic Equation

Understanding the standard form of a quadratic equation is crucial. This form is represented as \[ax^{2} + bx + c = 0\]. Here, 'a', 'b', and 'c' are constants, and 'a' cannot be zero. The term 'ax^{2}' shows its highest degree is 2. This sets it apart from other types of equations. The presence of 'x^{2}' makes the curve of its graph a parabola. This foundation helps identify and work with quadratic equations.

Linear vs. Quadratic Equations

Linear equations and quadratic equations differ mainly in the highest power of their variable. A linear equation is usually in the form \[ax + b = 0\], where 'a' and 'b' are constants, and the highest power of 'x' is 1. The graph of a linear equation is a straight line. On the other hand, a quadratic equation has the highest power of 2, making the graph a parabola. This distinction is key when solving or graphing these equations.

Identifying Quadratic Equations

To identify whether an equation is quadratic, look for the characteristic term 'x^{2}'. Let’s apply this to each given option:
\

• Option a) \[ \pi x^{2} - \sqrt{5} x - 1 = 0\]: This fits the standard quadratic form, with 'a' as \pi, 'b' as -\sqrt{5}, and 'c' as -1.
• Option b) \[ 3 x^{2} - 1 = 0\]: Even without 'bx', it remains quadratic because of the 'x^{2}'.
• Option c) \[ 4 x + 5 = 0\]: This is linear. It lacks 'x^{2}', making it non-quadratic.
• Option d) \[ 0.009 x^{2} = 0\]: Despite missing 'bx' and 'c', the 'x^{2}' term means it is quadratic.

This process quickly distinguishes quadratic equations from others.