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blagie [28]
11 months ago
10

Dentify on which quadratic function is positive.

Mathematics
2 answers:
Olin [163]11 months ago
4 0

Answer:

\textsf{$y = 2x^2 - 17x + 30$: \quad $\left(-\infty, \dfrac{5}{2}\right) \cup (6, \infty)$}

\textsf{$y = - x^2 - 6x - 8$: \quad $\left(-\infty, -4\right) \cup (-2, \infty)$}

Step-by-step explanation:

A function is positive when it is <u>above the x-axis</u>, and negative when it is <u>below the x-axis</u>.

---------------------------------------------------------------------------------

<u>Given quadratic equation</u>:

y = 2x^2 - 17x + 30

Factor the equation:

\implies y = 2x^2 - 17x + 30

\implies y = 2x^2 - 5x-12x + 30

\implies y=x(2x-5)-6(2x-5)

\implies y=(x-6)(2x-5)

The x-intercepts of the parabola are when y = 0.

To find the <u>x-intercepts</u>, set each factor equal to zero and solve for x:

\implies x-6=0 \implies x=6

\implies 2x-5=0 \implies x=\dfrac{5}{2}

Therefore, the x-intercepts are x = ⁵/₂ and x = 6.

The leading coefficient of the given function is positive, so the <u>parabola opens upwards</u>.  

The function is positive when it is <u>above the x-axis</u>.

Therefore, the function is positive for the values of x less than the smallest x-intercept and more than the largest x-intercept:

  • \textsf{Solution: \quad $x < \dfrac{5}{2}$ \;and \;$x > 6$}
  • \textsf{Interval notation: \quad $\left(-\infty, \dfrac{5}{2}\right) \cup (6, \infty)$}

---------------------------------------------------------------------------------

<u>Given quadratic equation</u>:

y = - x^2 - 6x - 8

Factor the equation:

\implies y = - x^2 - 6x - 8

\implies y = -(x^2 +6x +8)

\implies y = -(x^2 +4x +2x+8)

\implies y = -((x(x+4)+2(x+4))

\implies y = -(x+4)(x+2)

The x-intercepts of the parabola are when y = 0.

To find the <u>x-intercepts</u>, set each factor equal to zero and solve for x:

\implies x+4=0 \implies x=-4

\implies x+2=0 \implies x=-2

Therefore, the x-intercepts are x = -4 and x = -2.

The leading coefficient of the given function is negative, so the <u>parabola opens downwards</u>.  

The function is negative when it is <u>below the x-axis</u>.

Therefore, the function is negative for the values of x less than the smallest x-intercept and more than the largest x-intercept:

  • \textsf{Solution: \quad $x < -4$ \;and \;$x > -2$}
  • \textsf{Interval notation: \quad $\left(-\infty, -4\right) \cup (-2, \infty)$}

Hoochie [10]11 months ago
3 0

Step-by-step explanation:

Let us identify which quadratic function is positive. Yeah, let's start.

Y = { \red{ \sf{2 {x}^{2}  - 17x + 30}}}

By using factorisation method,

{ \red{ \sf{2 {x}^{2}  - 12x - 5x + 30}}}

Take common factors

{ \red{ \sf{2x(x - 6) - 5(x - 6)}}}

{ \red{ \sf{(2x - 5)}}} \:  \:  \:  \:  \:   \:  \: ||  \:  \:  \:  \:  \: { \red{ \sf{(x - 6)}}}

{ \red{ \sf{2x - 5 = 0}}} \:  \:  ||  \:  \: { \red{ \sf{x - 6 = 0}}}

{ \red{ \sf{2x = 5}}} \:  \:  \:  \:  \:  \:  \:    \:  \:  ||  \:  \:  \:  \: { \red{ \boxed{ \green{ \sf{x = 6}}}}}

{ \red{ \sf{{ \frac{ \cancel2}{ \cancel2}x}}}} = { \red{ \sf{ \frac{5}{2}}}}

{ \red{ \boxed{ \green{ \sf{x =  \frac{5}{2}}}}}}

____________________________________

Y = { \blue{ \sf {{ - x}^{2}  - 6x - 8}}}

By using factorisation method,

{ \blue{ \sf{ -  {x}^{2}  - 2x - 4x - 8}}}

Take common factors

{ \blue{ \sf{ - x(x + 2) - 4(x + 2)}}}

{ \blue{ \sf{( - x - 4)}}} \:  \:  \:  \:  \:  ||  \:  \:  \:  \:  \: { \blue{ \sf{(x + 2)}}}

{ \blue{ \sf{- x - 4 = 0}}} \:  \:  \: \: \: || \:  \:  \:  \: \: { \blue{ \sf{x + 2 = 0}}}

{ \blue{ \boxed{ \green{ \sf{x = -4}}}}} \: \: \: \: \: || \: \: \: \: \: { \blue{ \boxed{ \green{ \sf{x = -2}}}}}

Hence, the first quadratic function is positive and second quadratic function is negative.

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