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Answer :
To find two consecutive negative integers whose product is 600, let's represent these integers as [tex]\( x \)[/tex] and [tex]\( x + 1 \)[/tex]. Therefore, we have:
[tex]\[ x(x + 1) = 600 \][/tex]
This equation expands to:
[tex]\[ x^2 + x = 600 \][/tex]
Rearrange the equation to:
[tex]\[ x^2 + x - 600 = 0 \][/tex]
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -600 \)[/tex].
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Plugging in the values, we calculate the discriminant:
[tex]\[ b^2 - 4ac = 1^2 - 4 \times 1 \times (-600) = 1 + 2400 = 2401 \][/tex]
The discriminant is positive, indicating that there are two real solutions. Now, calculate the two possible values for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-1 \pm \sqrt{2401}}{2} \][/tex]
Since [tex]\( \sqrt{2401} = 49 \)[/tex], the solutions are:
[tex]\[ x = \frac{-1 + 49}{2} = 24 \][/tex]
[tex]\[ x = \frac{-1 - 49}{2} = -25 \][/tex]
Therefore, the two consecutive negative integers are [tex]\( -25 \)[/tex] and [tex]\( -24 \)[/tex]. The lesser of these two is [tex]\( -25 \)[/tex].
[tex]\[ x(x + 1) = 600 \][/tex]
This equation expands to:
[tex]\[ x^2 + x = 600 \][/tex]
Rearrange the equation to:
[tex]\[ x^2 + x - 600 = 0 \][/tex]
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -600 \)[/tex].
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Plugging in the values, we calculate the discriminant:
[tex]\[ b^2 - 4ac = 1^2 - 4 \times 1 \times (-600) = 1 + 2400 = 2401 \][/tex]
The discriminant is positive, indicating that there are two real solutions. Now, calculate the two possible values for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-1 \pm \sqrt{2401}}{2} \][/tex]
Since [tex]\( \sqrt{2401} = 49 \)[/tex], the solutions are:
[tex]\[ x = \frac{-1 + 49}{2} = 24 \][/tex]
[tex]\[ x = \frac{-1 - 49}{2} = -25 \][/tex]
Therefore, the two consecutive negative integers are [tex]\( -25 \)[/tex] and [tex]\( -24 \)[/tex]. The lesser of these two is [tex]\( -25 \)[/tex].
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