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Sagot :
To find a pair of integers whose product is -48 and whose sum is -8, let's denote these integers as [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
1. Identify the equations:
- The product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is -48:
[tex]\[ x \times y = -48 \][/tex]
- The sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is -8:
[tex]\[ x + y = -8 \][/tex]
2. Set up a system of equations:
[tex]\[ \begin{cases} x \times y = -48 \\ x + y = -8 \end{cases} \][/tex]
3. Solve for [tex]\( y \)[/tex]:
- From the sum equation, express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = -8 - x \][/tex]
4. Substitute [tex]\( y \)[/tex] in the product equation:
[tex]\[ x \times (-8 - x) = -48 \][/tex]
- Simplify the equation:
[tex]\[ -8x - x^2 = -48 \][/tex]
- Rearrange the equation:
[tex]\[ x^2 + 8x - 48 = 0 \][/tex]
5. Solve the quadratic equation:
- The quadratic equation is in the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here [tex]\( a = 1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = -48 \)[/tex].
- The quadratic formula to find the roots is:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
- Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
[tex]\[ x = \frac{{-8 \pm \sqrt{{8^2 - 4 \cdot 1 \cdot (-48)}}}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{{-8 \pm \sqrt{{64 + 192}}}}{2} \][/tex]
[tex]\[ x = \frac{{-8 \pm \sqrt{{256}}}}{2} \][/tex]
[tex]\[ x = \frac{{-8 \pm 16}}{2} \][/tex]
6. Calculate the roots:
- For the first root:
[tex]\[ x = \frac{{-8 + 16}}{2} \][/tex]
[tex]\[ x = \frac{{8}}{2} \][/tex]
[tex]\[ x = 4 \][/tex]
- For the second root:
[tex]\[ x = \frac{{-8 - 16}}{2} \][/tex]
[tex]\[ x = \frac{{-24}}{2} \][/tex]
[tex]\[ x = -12 \][/tex]
7. Find corresponding [tex]\( y \)[/tex] values:
- If [tex]\( x = 4 \)[/tex], then:
[tex]\[ y = -8 - 4 \][/tex]
[tex]\[ y = -12 \][/tex]
- If [tex]\( x = -12 \)[/tex], then:
[tex]\[ y = -8 - (-12) \][/tex]
[tex]\[ y = 4 \][/tex]
8. Conclusion:
- The pairs [tex]\((x, y)\)[/tex] that satisfy the given conditions are:
[tex]\((-12, 4)\)[/tex] and [tex]\( (4, -12)\)[/tex].
So, the integers whose product is -48 and whose sum is -8 are [tex]\((-12, 4)\)[/tex] or [tex]\((4, -12)\)[/tex].
1. Identify the equations:
- The product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is -48:
[tex]\[ x \times y = -48 \][/tex]
- The sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is -8:
[tex]\[ x + y = -8 \][/tex]
2. Set up a system of equations:
[tex]\[ \begin{cases} x \times y = -48 \\ x + y = -8 \end{cases} \][/tex]
3. Solve for [tex]\( y \)[/tex]:
- From the sum equation, express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = -8 - x \][/tex]
4. Substitute [tex]\( y \)[/tex] in the product equation:
[tex]\[ x \times (-8 - x) = -48 \][/tex]
- Simplify the equation:
[tex]\[ -8x - x^2 = -48 \][/tex]
- Rearrange the equation:
[tex]\[ x^2 + 8x - 48 = 0 \][/tex]
5. Solve the quadratic equation:
- The quadratic equation is in the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here [tex]\( a = 1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = -48 \)[/tex].
- The quadratic formula to find the roots is:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
- Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
[tex]\[ x = \frac{{-8 \pm \sqrt{{8^2 - 4 \cdot 1 \cdot (-48)}}}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{{-8 \pm \sqrt{{64 + 192}}}}{2} \][/tex]
[tex]\[ x = \frac{{-8 \pm \sqrt{{256}}}}{2} \][/tex]
[tex]\[ x = \frac{{-8 \pm 16}}{2} \][/tex]
6. Calculate the roots:
- For the first root:
[tex]\[ x = \frac{{-8 + 16}}{2} \][/tex]
[tex]\[ x = \frac{{8}}{2} \][/tex]
[tex]\[ x = 4 \][/tex]
- For the second root:
[tex]\[ x = \frac{{-8 - 16}}{2} \][/tex]
[tex]\[ x = \frac{{-24}}{2} \][/tex]
[tex]\[ x = -12 \][/tex]
7. Find corresponding [tex]\( y \)[/tex] values:
- If [tex]\( x = 4 \)[/tex], then:
[tex]\[ y = -8 - 4 \][/tex]
[tex]\[ y = -12 \][/tex]
- If [tex]\( x = -12 \)[/tex], then:
[tex]\[ y = -8 - (-12) \][/tex]
[tex]\[ y = 4 \][/tex]
8. Conclusion:
- The pairs [tex]\((x, y)\)[/tex] that satisfy the given conditions are:
[tex]\((-12, 4)\)[/tex] and [tex]\( (4, -12)\)[/tex].
So, the integers whose product is -48 and whose sum is -8 are [tex]\((-12, 4)\)[/tex] or [tex]\((4, -12)\)[/tex].
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