Question

Find the zeros of the function. \(g(x)=7 x^2+21\)

Short answer

The function \(g(x) = 7x^2 + 21\) does not have real zeros because its discriminant is negative.

Step-by-step solution

Set up the Equation

Start by setting the given function \(g(x) = 7x^2 + 21\) equal to zero: \[0 = 7x^2+21.\] This equation will help find the values of x that make the function equals zero.

Solve for \(x\)

Solve the above equation for \(x\). First, isolate the \(x^2\) term by subtracting 21 from both sides and dividing by 7: \[-3 = x^2.\] Then try to solve for \(x\) by taking the square root of both sides. Here, you'll find that this equation does not have real roots because the square root of a negative number is undefined in the real number system.

Identify the lack of Real Roots

As there can't be a real square root of a negative number, conclude that the function \(g(x) = 7x^2 + 21\) has no real zeros.

Key concepts

Solving Quadratic Equations

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

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.

Complex Numbers

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.