Solve x²+ ½x - 4½: Finding Values Where Function is Positive

Look at the function below:

y=x2+12x412 y=x^2+\frac{1}{2}x-4\frac{1}{2}

Then determine for which values of x x the following is true:

f(x)>0 f(x) > 0

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Step-by-step written solution

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1

Understand the problem

Look at the function below:

y=x2+12x412 y=x^2+\frac{1}{2}x-4\frac{1}{2}

Then determine for which values of x x the following is true:

f(x)>0 f(x) > 0

2

Step-by-step solution

To solve the problem, let's determine when y=x2+12x4.5 y = x^2 + \frac{1}{2}x - 4.5 is greater than zero by following these steps:

  • Step 1: Find the roots of the quadratic equation using the quadratic formula.
  • Step 2: Determine the intervals defined by these roots.
  • Step 3: Identify where the quadratic function is positive.

**Step 1**: Given the quadratic function y=x2+12x4.5 y = x^2 + \frac{1}{2}x - 4.5 , the coefficients are a=1 a = 1 , b=12 b = \frac{1}{2} , and c=4.5 c = -4.5 . Apply the quadratic formula:

x=b±b24ac2a=(12)±(12)24(1)(4.5)2(1) x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-\left(\frac{1}{2}\right) \pm \sqrt{\left(\frac{1}{2}\right)^2 - 4(1)(-4.5)}}{2(1)}

Simplify further:

x=12±14+182=12±7342 x = \frac{-\frac{1}{2} \pm \sqrt{\frac{1}{4} + 18}}{2} = \frac{-\frac{1}{2} \pm \sqrt{\frac{73}{4}}}{2}

This becomes:

x=1±734 x = \frac{-1 \pm \sqrt{73}}{4}

**Step 2**: The roots are x1=1734 x_1 = \frac{-1 - \sqrt{73}}{4} and x2=1+734 x_2 = \frac{-1 + \sqrt{73}}{4} . The function changes sign at the roots. Since the quadratic opens upwards (as a=1>0 a = 1 > 0 ), it will be positive between the roots:

**Step 3**: Identify the interval where the quadratic is positive:

1734<x<1+734 \frac{-1 - \sqrt{73}}{4} < x < \frac{-1 + \sqrt{73}}{4}

Therefore, the values of x x for which the function is greater than zero are within this interval.

The correct solution is 1734<x<1+734 \frac{-1 - \sqrt{73}}{4} < x < \frac{-1 + \sqrt{73}}{4} .

3

Final Answer

1734<x<1+734 \frac{-1-\sqrt{73}}{4} < x < \frac{-1+\sqrt{73}}{4}

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of \( x \) where \( f\left(x\right) > 0 \).

AAABBBCCCX

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