the function $g(x)$ is defined as $g(x)=6x^{2}+23x - 4$. when does $g(x)=0$?\n$x=-6$ or $x =…

the function $g(x)$ is defined as $g(x)=6x^{2}+23x - 4$. when does $g(x)=0$?\n$x=-6$ or $x = \frac{1}{4}$\n$x=-4$ or $x=\frac{1}{6}$\n$x=-\frac{1}{4}$ or $x = 6$\n$x=-\frac{1}{6}$ or $x = 4$

the function $g(x)$ is defined as $g(x)=6x^{2}+23x - 4$. when does $g(x)=0$?\n$x=-6$ or $x = \frac{1}{4}$\n$x=-4$ or $x=\frac{1}{6}$\n$x=-\frac{1}{4}$ or $x = 6$\n$x=-\frac{1}{6}$ or $x = 4$

Answer

Explanation:

Step1: Set the function equal to 0

$6x^{2}+23x - 4=0$

Step2: Factor the quadratic equation

We need to find two numbers that multiply to $6\times(- 4)=-24$ and add up to 23. The numbers are 24 and - 1. Rewrite the middle - term: $6x^{2}+24x-x - 4 = 0$. Group the terms: $(6x^{2}+24x)-(x + 4)=0$. Factor out the common factors from each group: $6x(x + 4)-(x + 4)=0$. Factor out $(x + 4)$: $(x + 4)(6x-1)=0$.

Step3: Use the zero - product property

If $(x + 4)(6x-1)=0$, then either $x+4 = 0$ or $6x-1=0$. For $x + 4=0$, we get $x=-4$. For $6x-1=0$, we solve for $x$: $6x=1$, so $x=\frac{1}{6}$.

Answer:

$x=-4$ or $x=\frac{1}{6}$