2. Between which two consecutive integers does $$ \sqrt{138} $$ lie? .3. Given $$ \mathrm{q}=\sqrt{b^{2}-4 a c} $$. If $$ \mathrm{a}=-1 ; \mathrm{b}=-2 $$ and $$ \mathrm{c}=4 $$. Determine between which two Consecutive integers does q lie.

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To find between which two consecutive integers $$ \sqrt{138} $$ lies, we need to find the square root of 138 and then determine the two consecutive integers it falls between.

Given:
$$ a = -1 $$
$$ b = -2 $$
$$ c = 4 $$

We are asked to find the value of $$ q = \sqrt{b^{2}-4ac} $$ and determine between which two consecutive integers it lies.

Let's calculate the value of $$ q $$ first.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\left(-2ight)^{2}-4\left(-1ight)	imes 4}$$
- step1: Multiply:
  $$\sqrt{\left(-2ight)^{2}+16}$$
- step2: Add the numbers:
  $$\sqrt{20}$$
- step3: Write as a product:
  $$\sqrt{4	imes 5}$$
- step4: Write in exponential form:
  $$\sqrt{2^{2}	imes 5}$$
- step5: Use the properties of radicals:
  $$\sqrt{2^{2}}	imes \sqrt{5}$$
- step6: Simplify the root:
  $$2\sqrt{5}$$
The value of $$ q $$ is approximately 4.472136.

Now, we need to find between which two consecutive integers $$ q $$ lies. Since $$ q $$ is approximately 4.472136, it lies between the integers 4 and 5.
$$ \sqrt{138} $$ lies between 11 and 12. The value of $$ q = \sqrt{b^{2}-4ac} $$ is approximately 4.472136, which lies between 4 and 5.

2. Between which two consecutive integers does \( \sqrt{138} \) lie? .3. Given \( \mathrm{q}=\sqrt{b^{2}-4 a c} \). If \( \mathrm{a}=-1 ; \mathrm{b}=-2 \) and \( \mathrm{c}=4 \). Determine between which two Consecutive integers does q lie.

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