Chapter 1.4, Problem 47E

Solution

To find: The zeros of the function $y=15x^2-5x-20$ by writing in intercept form.

  $x=-1$ , $x=\frac{3}{4}$ .

Given information:

The function, $y=15x^2-5x-20$ .

Concept used:

The intercept form of a quadratic function is given as $y=\left(x-a\right)\left(x-b\right)$ , where $a$ and $b$ are zeros of the function.

Calculation:

Rewrite the function in intercept form as follows:

  $\begin{array}{c}y=15x^2-5x-20\\ =5\left(3x^2-x-4\right)\\ =5\left(\left(3x^2+3x\right)+\left(-4x-4\right)\right)\\ =5\left(x\left(x+1\right)-4\left(x+1\right)\right)\\ =5\left(x+1\right)\left(3x-4\right)\end{array}$

The intercept of the function are the points where the factors $x+1=0$ and $3x-4=0$ .

Solve the two equations to find the intercepts.

  $\begin{array}{c}x+1=0\\ x=-1\end{array}$

And,

  $\begin{array}{c}3x-4=0\\ x=\frac{3}{4}\end{array}$

Conclusion:

The zeros of the function are $x=-1$ and $x=\frac{3}{4}$ .