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The zeros of a function represent the points where the function’s value is zero. For a quadratic function like \( h(x) = 4x^2 + 36 \), these are the values of \( x \) that result in \( h(x) = 0 \). These are also known as roots or solutions of the equation. Finding the zeros of a quadratic function involves solving the equation for when the output is zero. If the function is set up in standard form as \( ax^2 + bx + c = 0 \), identifying the zeros means determining which values make this equation hold true. In our case, the function is \( 4x^2 + 36 \), and we're looking to solve the equation \( 4x^2 + 36 = 0 \). This leads us to use algebraic methods like the Quadratic Formula to find the zeros, as it works universally for quadratic equations. We perform operations to isolate and solve for \( x \), thus finding the zeros.

The Quadratic Formula is a powerful tool to solve any quadratic equation, which is expressed in the standard form \( ax^2 + bx + c = 0 \). This formula is written as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In the problem we are considering, the given quadratic function \( h(x) = 4x^2 + 36 \) can be solved using the Quadratic Formula. We identify \( a = 4 \), \( b = 0 \), and \( c = 36 \). By plugging these values into the formula, we attempt to find the zeros of the function. In this particular case, however, since \( b^2 - 4ac = -576 \), we encounter a situation where the term under the square root (the discriminant) is negative. This scenario indicates that our equation's solutions are not real numbers, which necessitates understanding of complex numbers.

When solving a quadratic equation, a negative discriminant indicates the presence of imaginary numbers as part of the solutions. Imaginary numbers arise from taking the square root of a negative number. In our problem \( 4x^2 + 36 = 0 \), the discriminant \(-576\) is negative, which means the equation has no real zeros.
Instead, the solutions involve imaginary numbers. Imaginary numbers are noted with \( i \), where \( i^2 = -1 \). This leads us to evaluate the square root of negative numbers as multiples of \( i \).

• So, \( \sqrt{-1} = i \)
• \( \sqrt{-576} = \sqrt{576} \times i = 24i \)

Thus, when using the Quadratic Formula, we adjust to accommodate this understanding, presenting our solutions using \( i \). For our equation, this results in solutions of \( x = \frac{ \pm 24i}{8} = \pm 3i \). This illustrates how imaginary numbers can be integral in finding solutions where no real zeros exist for a quadratic function.