Question

Find the zero(s) of the function. $$ g(x)=x^2+14 x+49 $$

Short answer

The function \(g(x) = x^2 + 14x + 49\) has exactly one zero at \(x = -7\).

Step-by-step solution

Write Down the Equation

To begin solving for the zeros, write down the function as an equation, setting it equal to zero: \(x^2 + 14x + 49 = 0\)

Apply the Quadratic Formula

Apply the quadratic formula to find the roots of the equation. The formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here \(a = 1\), \(b = 14\), and \(c = 49\). Again, the discriminant that is calculated as part of the quadratic formula (\(b^2 - 4ac\)) determines the number and types of roots the equation has. In this case, the discriminant is \(14^2 - 4*1*49 = 0\). Because the discriminant is equal to zero, this equation has exactly one real root.

Solve for the Zero

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula. The equation becomes \(x = \frac{-14 \pm \sqrt{0}}{2*1}\), which simplifies to \(-7\).

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool in algebra that allows us to find the solutions to quadratic equations, which are equations of the type \( ax^2 + bx + c = 0 \). When you encounter such an equation, you can quickly apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula provides the solution by taking into account the coefficients \(a\), \(b\), and \(c\) of the quadratic equation. To use the formula, you simply plug in the values for \(a\), \(b\), and \(c\) from your equation, and calculate. It will give you the zeros of the quadratic function, which are the points where the graph of the function crosses the x-axis.

If we consider the given function \( g(x)=x^2+14x+49 \), it's evident that \(a=1\), \(b=14\), and \(c=49\). Applying the quadratic formula, you can compute the zeroes of the function.

Discriminant

The discriminant is the part of the quadratic formula under the square root: \( b^2 - 4ac \). It plays a crucial role in determining the nature and number of roots of a quadratic equation. The discriminant can tell us whether the roots are:

• Real and distinct (if \( b^2 - 4ac > 0 \))
• Real and identical (if \( b^2 - 4ac = 0 \))
• Complex or non-real (if \( b^2 - 4ac < 0 \))

For the quadratic equation \(x^2 + 14x + 49 = 0\), the discriminant is \(14^2 - 4\cdot1\cdot49 = 0\), indicating that there is exactly one real root. The root is repeated, which implies that the graph of the quadratic function touches the x-axis at one point only, visually representing this unique solution.

Roots of an Equation

When we speak of the roots of an equation, we refer to the solution(s) to the equation set to zero. In the case of quadratic equations, these roots are the values of \(x\) that make the equation \(ax^2+bx+c=0\) true. These are also the x-coordinates where the graph of the quadratic function intersects the x-axis.

In our example, with the quadratic function \(g(x)=x^2+14x+49\), by using the quadratic formula and finding its discriminant to be zero, we have determined that the function has a unique root. This root is \(x=-7\). It denotes the point on the graph where it touches the x-axis, and this point is called the vertex of the parabola when the discriminant is zero. It's important to understand that roots are specific values that satisfy the equation; in the context of functions, they offer a graphical point of reference for the intersection with the axis.