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The function
k is given by
k(x)=3x^(2)-19 x-14. Find all values of
x for which
k(x) > 0.

The function $k$ is given by $k(x)=3 x^{2}-19 x-14$ . Find all values of $x$ for which $k(x)>0$ .

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Domain and range of quadratic functions: equations

Full solution

Q. The function

k

is given by

k(x)=3 x^{2}-19 x-14

. Find all values of

x

for which

k(x)>0

.

Factor quadratic equation: Find the roots of the quadratic equation by factoring $k(x) = 3x^2 - 19x - 14$ . $k(x) = (3x + 1)(x - 14)$ Set each factor equal to zero to find the roots. $3x + 1 = 0$ and $x - 14 = 0$
Solve for roots: Solve for $x$ in each equation. $\newline$ For $3x + 1 = 0$ , $x = -\frac{1}{3}$ $\newline$ For $x - 14 = 0$ , $x = 14$
Determine intervals to test: Determine the intervals to test around the roots. $\newline$ We have two intervals: $x < -\frac{1}{3}$ , $-\frac{1}{3} < x < 14$ , and $x > 14$ .
Choose test points: Choose test points from each interval and plug them into $k(x)$ . For $x < -\frac{1}{3}$ , use $x = -1$ . For $-\frac{1}{3} < x < 14$ , use $x = 0$ . For $x > 14$ , use $x = 15$ .
Evaluate $k(x)$ : Evaluate $k(x)$ at each test point. $\newline$ $k(-1) = 3(-1)^2 - 19(-1) - 14 = 3 + 19 - 14 = 8$ $\newline$ $k(0) = 3(0)^2 - 19(0) - 14 = -14$ $\newline$ $k(15) = 3(15)^2 - 19(15) - 14 = 675 - 285 - 14 = 376$
Determine where $k(x) > 0$ : Determine where $k(x)$ is greater than $0$ based on the test points. $\newline$ $k(x) > 0$ when $x < -\frac{1}{3}$ and when $x > 14$ .

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