Question

Find the zero(s) of the function. (See Example 4.) \(g(x)=x^2+11 x\)

Short answer

The zeros of the function \(g(x)=x^2+11 x\) are \(x = 0\) and \(x = -11\).

Step-by-step solution

Rewriting the Function

1: Start by rewriting \(g(x)\) as: \[x^2 + 11x = 0 \]

Factoring the Function

2: Factor out an 'x' from each term, this leaves: \[x (x + 11 ) = 0 \]

Setting Each Factor Equal to Zero

3: Next set each factor equal to zero and solve to derive the roots/zero(s) of function: \[x = 0 \]and \[x + 11 = 0\]

Solving for Zero(s)

4: After solving for x, the answers are \[x = 0\] and the other root is \[x = -11\]

Key concepts

Factoring Quadratics

Factoring quadratics is a crucial skill in solving quadratic equations. In simple terms, it involves expressing a quadratic function as a product of its factors. Think of factoring as the reverse of expanding. You start with the expanded form, like a trinomial, and aim to rewrite it in its factored form.
For the function \(g(x) = x^2 + 11x\), we can observe that both terms share a common factor, which is \(x\). By factoring out \(x\) from each term, you simplify the expression to \(x(x + 11)\).

• Identify any common factors in your expression.
• Rewrite the quadratic equation using these factors to simplify it.

These factors are crucial because they set up the next part of solving quadratic equations. Factoring makes it much easier to find solutions as it clearly lays out the terms that we can further manipulate.

Solving Quadratic Equations

Once you have factored the quadratic equation, the next step is finding the solutions or 'roots.' These are the values of \(x\) that make the equation true, meaning they make the expression equal zero when substituted in.
For instance, with the factored form of \(x(x + 11) = 0\), we have broken down what seemed like a complex equation into simpler linear pieces.

• Write the equation in a factored form.
• Set each factor individually equal to zero.
• Solve those simple equations to find the values of \(x\).

This step clarifies the values for \(x\) that solve the original quadratic expression. Each solution is also called a root or zero of the function, and these determine where the curve of the function intersects the x-axis.

Zero-Product Property

The zero-product property is a straightforward, yet powerful tool used in solving quadratic equations. It states that if the product of multiple factors is zero, at least one of those factors must be zero. This is the principle we apply when working with factored quadratic equations.
With \(x(x + 11) = 0\), according to the zero-product property, either \(x = 0\) or \(x + 11 = 0\). It means we have simplified our task to solving two linear equations:

• \(x = 0\), giving us the first solution.
• Solving \(x + 11 = 0\) leads to \(x = -11\), giving us the second solution.

Using this property is fundamental to easily solving factored quadratic equations. It highlights how we can convert a complicated task into a more manageable series of steps, empowering us to tackle quadratics with confidence.