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Answer :
To solve the problem of finding two consecutive negative integers whose product is 600, let's break it down step by step:
1. Understand the Problem:
- We need two consecutive negative integers. Let's represent the first (lesser) integer as [tex]\( x \)[/tex].
- The next consecutive integer will be [tex]\( x + 1 \)[/tex].
2. Set Up the Equation:
- The product of these integers is given as 600. Therefore, we can write the equation as:
[tex]\[
x \times (x + 1) = 600
\][/tex]
3. Simplify the Equation:
- Distribute [tex]\( x \)[/tex] in the equation:
[tex]\[
x^2 + x = 600
\][/tex]
- Rearrange it into a standard form of a quadratic equation:
[tex]\[
x^2 + x - 600 = 0
\][/tex]
4. Solve the Quadratic Equation:
- You can use the quadratic formula to solve for [tex]\( x \)[/tex], which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -600 \)[/tex]. Substitute these values into the formula:
[tex]\[
x = \frac{-1 \pm \sqrt{1 + 2400}}{2}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{2401}}{2}
\][/tex]
- [tex]\(\sqrt{2401}\)[/tex] gives us 49. Hence:
[tex]\[
x = \frac{-1 + 49}{2} \quad \text{or} \quad x = \frac{-1 - 49}{2}
\][/tex]
[tex]\[
x = 24 \quad \text{or} \quad x = -25
\][/tex]
5. Choose the Negative Solution:
- Since we are looking for negative integers, we select the negative solution, which is [tex]\( x = -25 \)[/tex].
6. Determine the Lesser Integer:
- The consecutive integers are [tex]\( -25 \)[/tex] and [tex]\( -24 \)[/tex]. Hence, the lesser integer is [tex]\(-25\)[/tex].
Therefore, the value of the lesser integer is [tex]\(-25\)[/tex].
1. Understand the Problem:
- We need two consecutive negative integers. Let's represent the first (lesser) integer as [tex]\( x \)[/tex].
- The next consecutive integer will be [tex]\( x + 1 \)[/tex].
2. Set Up the Equation:
- The product of these integers is given as 600. Therefore, we can write the equation as:
[tex]\[
x \times (x + 1) = 600
\][/tex]
3. Simplify the Equation:
- Distribute [tex]\( x \)[/tex] in the equation:
[tex]\[
x^2 + x = 600
\][/tex]
- Rearrange it into a standard form of a quadratic equation:
[tex]\[
x^2 + x - 600 = 0
\][/tex]
4. Solve the Quadratic Equation:
- You can use the quadratic formula to solve for [tex]\( x \)[/tex], which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -600 \)[/tex]. Substitute these values into the formula:
[tex]\[
x = \frac{-1 \pm \sqrt{1 + 2400}}{2}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{2401}}{2}
\][/tex]
- [tex]\(\sqrt{2401}\)[/tex] gives us 49. Hence:
[tex]\[
x = \frac{-1 + 49}{2} \quad \text{or} \quad x = \frac{-1 - 49}{2}
\][/tex]
[tex]\[
x = 24 \quad \text{or} \quad x = -25
\][/tex]
5. Choose the Negative Solution:
- Since we are looking for negative integers, we select the negative solution, which is [tex]\( x = -25 \)[/tex].
6. Determine the Lesser Integer:
- The consecutive integers are [tex]\( -25 \)[/tex] and [tex]\( -24 \)[/tex]. Hence, the lesser integer is [tex]\(-25\)[/tex].
Therefore, the value of the lesser integer is [tex]\(-25\)[/tex].
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