Solve y = x² - 4x - 4: Finding Values Where Function is Positive

Look at the function below:

y=x24x4 y=x^2-4x-4

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

Follow each step carefully to understand the complete solution
1

Understand the problem

Look at the function below:

y=x24x4 y=x^2-4x-4

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 this problem, we will apply the following steps:

  • Step 1: Find the roots of the given quadratic function.
  • Step 2: Determine the sign of the quadratic between and beyond the roots.
  • Step 3: Conclude which values of x x make the quadratic positive.

Let's start with Step 1: Find the roots of the function y=x24x4 y = x^2 - 4x - 4 .
We use the quadratic formula: x=b±b24ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substituting a=1,b=4,c=4 a = 1, b = -4, c = -4 , we get:
Δ=b24ac=(4)24(1)(4)=16+16=32\Delta = b^2 - 4ac = (-4)^2 - 4(1)(-4) = 16 + 16 = 32

Thus, the roots are:
x=(4)±322(1)=4±322 x = \frac{-(-4) \pm \sqrt{32}}{2(1)} = \frac{4 \pm \sqrt{32}}{2}

Simplifying further gives us 32=42 \sqrt{32} = 4\sqrt{2} , so:
x=4±422=2±22 x = \frac{4 \pm 4\sqrt{2}}{2} = 2 \pm 2\sqrt{2}

The roots are x=2+22 x = 2 + 2\sqrt{2} and x=222 x = 2 - 2\sqrt{2} .

Step 2: Determine the sign of the quadratic.
Since the parabola opens upward (coefficient of x2 x^2 is positive), it is below the x-axis between the roots and above the x-axis outside the roots.

Step 3: Conclude values for which f(x)>0 f(x) > 0 .
f(x)>0 f(x) > 0 for x<222 x < 2 - 2\sqrt{2} or x>2+22 x > 2 + 2\sqrt{2} .

Finally, the solution to the problem is: x>2+22 x > 2 + 2\sqrt{2} or x<222 x < 2 - 2\sqrt{2} .

3

Final Answer

x>2+22 x > 2+2\sqrt{2} or x<222 x < 2-2\sqrt{2}

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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