Understanding Quadratic Equations

A quadratic equation is a polynomial equation where the highest power (degree) of the variable is 2. The standard form is \(ax^2 + bx + c = 0\), where \(a \neq 0\). We need to find the option where the variable '\(x\)' has a maximum exponent of 2.

Analyzing the Options

Let's examine each option to determine the highest power of '\(x\)':

• Option 1: \(x^2+3x^3+5\)

  The highest power of '\(x\)' is 3 (from the term \(3x^3\)). This is a cubic equation, not quadratic.

• Option 2: \(x^{3/5}+y^2-3\)

  This equation contains a fractional exponent (\(x^{3/5}\)). Quadratic equations must have integer exponents, specifically a highest exponent of 2.

• Option 3: \(5x^2+4-7x\)

  The terms are \(5x^2\), $4$ (which is \(4x^0\)), and \(-7x\) (which is \(-7x^1\)). The highest power of '\(x\)' is 2. This equation can be written in standard form as \(5x^2 - 7x + 4 = 0\). Therefore, this is a quadratic equation.

• Option 4: \(x^2-2x^{1/5}+2\)

  This equation includes a fractional exponent (\(x^{1/5}\)), disqualifying it as a quadratic equation.

• Option 5:

  This option is empty and cannot be evaluated.

Identifying the Quadratic Equation

Based on the analysis, the only equation that meets the definition of a quadratic equation (highest power of \(x\) is 2) is Option 3.