Question

for quadratic function $f$, the solutions to the equation $f(x)=0$ are $x=\frac{7}{5}$ and $x=-\frac{2}{3}$. which function could represent $f$?
a $f(x)=14x^2 + 11x - 15$
b $f(x)=15x^2 - 11x - 14$
c $f(x)=15x^2 + 11x - 14$
d $f(x)=14x^2 - 11x - 15$

Explanation:

Step1: Use roots to form factored function

If $x=\frac{7}{5}$ and $x=-\frac{2}{3}$ are solutions, then $(5x-7)$ and $(3x+2)$ are factors. The quadratic function is $f(x)=k(5x-7)(3x+2)$ where $k$ is a constant.

Step2: Expand the factored form

First multiply the binomials:
$$(5x-7)(3x+2)=5x\cdot3x + 5x\cdot2 -7\cdot3x -7\cdot2=15x^2+10x-21x-14=15x^2-11x-14$$
With $k=1$, this matches one of the options.

Step3: Verify with given options

Compare the expanded form to the choices; it matches option B.

Answer:

B. $f(x) = 15x^2 - 11x - 14$