Question

Find the zero(s) of the function. (See Example 4.) \(f(x)=4 x^2-12 x+9\)

Short answer

The function \(f(x) = 4x^2 -12x + 9\) has one zero at \(x = 1.5\).

Step-by-step solution

Identify a, b and c

The equation has the form \(ax^2 + bx + c = 0\). From the given function \(f(x) = 4x^2 - 12x + 9 \), we identify \(a = 4, b = -12, c = 9\).

Apply the Quadratic Formula

Applying the Quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), Here, \(x = \frac{--12 \pm \sqrt{(-12)^2 - 4*4*9}}{2*4} = \frac{12 \pm \sqrt{144 - 144}}{8}\).

Simplify

This simplifies to \(x = \frac{12 \pm 0}{8} = \frac{12}{8} = 1.5\). Note that the '±' in the formula would usually give two solutions, but since the value under the square root (∆, the discriminant) is zero here, we only get one solution.

Key concepts

Quadratic Formula

The Quadratic Formula is a reliable method for finding the solutions of any quadratic equation of the form \(ax^2+bx+c=0\). This formula provides the values of \(x\) that satisfy the equation, commonly referred to as the zeros or roots. The formula is expressed as:

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

Here, \(a\), \(b\), and \(c\) are constants derived from the quadratic equation. The \('±'\) symbol in the equation indicates two potential solutions.
First, determine the values of \(a\), \(b\), and \(c\). In the context of our example with the function \(f(x)=4x^2-12x+9\), \(a=4\), \(b=-12\), and \(c=9\). Using these values, calculate the solutions for \(x\) by substituting them into the Quadratic Formula.
This method solves for \(x\) by determining whether the equation has one real solution, two real solutions, or even complex solutions depending on the outcome of the discriminant part under the square root.

Discriminant

The discriminant is a crucial component within the Quadratic Formula that indicates the nature and number of solutions a quadratic equation might possess. The discriminant, \(\Delta\), is the part of the formula \(b^2-4ac\), found beneath the square root.
The value of the discriminant determines the following:

• If \(\Delta > 0\), the quadratic equation has two distinct real solutions.
• If \(\Delta = 0\), the quadratic equation has exactly one real solution, also known as a repeated root.
• If \(\Delta < 0\), the equation has no real solutions, but two complex solutions.

In the given function \(f(x) = 4x^2 - 12x + 9\), the discriminant becomes \((−12)^2 − 4 \cdot 4 \cdot 9 = 144 - 144 = 0\).
Since the discriminant equals zero, this confirms that the quadratic equation has exactly one real solution.

Finding Zeros

Finding zeros of a quadratic function means determining the values of \(x\) that make the function equal to zero. These zeros are where the function graph intersects the x-axis. For the formula \(f(x) = ax^2+bx+c\), the task is to solve for \(x\) when \(f(x)=0\).
By using the Quadratic Formula, we substitute \(a\), \(b\), and \(c\) from the quadratic equation into the formula to solve for \(x\). In our specific function \(f(x) = 4x^2 - 12x + 9\), the Quadratic Formula was used, resulting in:

• \(\frac{12 \pm \sqrt{144 - 144}}{8} = \frac{12 \pm 0}{8} = \frac{12}{8} = 1.5\)

With the discriminant equal to zero, we find that there is one zero, \(x = 1.5\). This means that the function's graph just touches the x-axis at \(x = 1.5\), signifying a single intersection at this point.