To solve the equation [tex]\( 2x^2 = 310 \)[/tex] and determine between which two consecutive negative integers a solution lies, we follow these steps:
1. First, solve the equation [tex]\( 2x^2 = 310 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
x^2 = \frac{310}{2} = 155
\][/tex]
2. Next, take the square root of both sides of the equation to find the solutions for [tex]\( x \)[/tex]:
[tex]\[
x = \pm \sqrt{155}
\][/tex]
The positive solution is [tex]\( \sqrt{155} \)[/tex] and the negative solution is [tex]\( -\sqrt{155} \)[/tex].
3. We are only interested in the negative solution, so we focus on [tex]\( -\sqrt{155} \)[/tex].
4. Estimate [tex]\( \sqrt{155} \)[/tex]:
- We know that [tex]\( 12^2 = 144 \)[/tex] and [tex]\( 13^2 = 169 \)[/tex].
- Thus, [tex]\( 12 < \sqrt{155} < 13 \)[/tex].
5. From the above estimation, we know that:
[tex]\[
-13 < -\sqrt{155} < -12
\][/tex]
Therefore, the negative solution [tex]\( -\sqrt{155} \)[/tex] lies between the two consecutive negative integers [tex]\( -13 \)[/tex] and [tex]\( -12 \)[/tex].
Hence, the correct answer is:
- (13, -12)
