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Understanding quadratic equations is crucial for finding the zeros of a function. A quadratic equation is typically presented in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \), are coefficients and \( x \) represents the unknown variable.

When attempting to find the solutions to a quadratic equation, one initially aims to isolate the \( x \) term. This process often includes moving terms to one side of the equation and factoring, or using the quadratic formula. However, the challenge arises when dealing with a special type of quadratic equations, such as \( 7x^2 + 21 = 0 \), which after simplification leads to \( x^2 = -3 \).

In this case, no real number multiplied by itself will yield a negative number. Therefore, simple factorization or rearrangement will not help to find real solutions. Advanced methods, including the concept of complex numbers, may then come into play.

Real roots of a function are the x-values at which the graph of the function intersects with the x-axis. In other words, they are the solutions to the function when it is set to zero. To find the real roots of a quadratic equation, one can try factoring, completing the square, or applying the quadratic formula.

For example, an equation like \( x^2 = 9 \) will have real roots because both \( 3 \) and \( -3 \) are real numbers that satisfy the equation. However, when you come across an equation such as \( x^2 = -3 \), a different approach is needed since the square root of a negative number is not a real number. This realization is pivotal, as it signifies that the function in question does not touch the x-axis, and thus, has no real zeros or roots.

To handle situations where the roots of a quadratic equation are not real, mathematicians introduced the concept of complex numbers. A complex number is formed with a real part and an imaginary part, typically written as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit defined as the square root of -1.

In the exercise \( g(x) = 7x^2 + 21 \), we found that solving the equation leads to \( x^2 = -3 \). The square root of -3 can be expressed using complex numbers as \( \sqrt{-3} = \sqrt{3}i \). Therefore, the zeros of \( g(x) \) can be expressed as complex numbers \( x = \sqrt{3}i \) and \( x = -\sqrt{3}i \) acknowledging the existence of two solutions, though not in the realm of real numbers.