Chapter 3.1, Problem 52E

To determine

To calculate: The zeros of the given function.

Answer to Problem 52E

The required solution is $x=\frac{3}{2}$ .

Explanation of Solution

Given information:

The function: $f(x)=4x^2-12x+9$

Calculation:

The given function is $f(x)=4x^2-12x+9$ .

To find the zeros of the given function, substitute 0 for $f\left(x\right)$ .

  $4x^2-12x+9=0$

To solve the given equation by factoring, split the middle term with respect to the product of the coefficient of $x^2$ and the constant term.

  $\begin{array}{c}4x^2-12x+9=0\\ 4x^2-6x-6x+9=0\end{array}$

Take the common factor 2x from the first two terms and $-3$ from the last two terms.

  $\begin{array}{c}4x^2-6x-6x+9=0\\ 2x\left(2x-3\right)-3\left(2x-3\right)=0\\ \left(2x-3\right)\left(2x-3\right)=0\end{array}$

Solve the above expression for x .

  $\begin{array}{c}2x-3=0\\ \frac{\cancel{2}x}{\cancel{2}}=\frac{3}{2}\\ x=\frac{3}{2}\end{array}$

And, $\begin{array}{c}2x-3=0\\ \frac{\cancel{2}x}{\cancel{2}}=\frac{3}{2}\\ x=\frac{3}{2}\end{array}$

Hence, the required solution is $x=\frac{3}{2}$ .