Question

The sum of two integers is -11 and their product is -80. What are the two integers?
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Answer 1

Given

• Sum of two integers = (-11)
• Product of two integer= (-80)

To Find

• The two integers

Let the Two integers be x and y

then

• $\sf \: x+y= -11 \: \: \: \: \: \: - - - - - (1)$
• $\sf \: x× y= -80 \: \: \: \: \: \: - - - - - (2)$

we know that

• $\boxed{ \bf{ {(a - b) }^{2} = {a}^{2} + {b}^{2} - 2ab}}$

${~~~~~:~~\implies \sf (x - y) ^{2} = {x}^{2} + {y}^{2} -2 xy}$

• $\boxed{ \bf{ {a}^{2} + {b}^{2} ={ (a + b)}^{2} - 2ab}}$

${~~~~~:~~\implies \sf (x - y) ^{2} = \{{ (x + y) }^{2} - 2xy \} -2 xy}$

• Putting ( x+y) =-11 and xy= -80

${~~~~~:~~\implies \sf (x - y) ^{2} = { (- 11) }^{2} - 2 \times ( - 80) -2 \times ( - 80)}$

${~~~~~:~~\implies \sf (x - y) ^{2} = 121 + 160 + 160}$

${~~~~~:~~\implies \sf (x - y) ^{2} = 121 + 320}$

${~~~~~:~~\implies \sf (x - y) ^{2} = 441}$

${~~~~~:~~\implies \sf x - y = \sqrt{441} }$

${~~~~~:~~\implies \sf x - y = \sqrt{3 \times 3 \times 7 \times 7} }$

${~~~~~:~~\implies \sf x - y = \sqrt{21 \times 21} }$

${~~~~~:~~\implies \sf x - y = \sqrt{ {21}^{2} } }$

${~~~~~:~~\implies \sf x - y = 21 \: \: \: \: \: \: - - - - (3)}$

Now,

• $\sf \: x - y = 21 \: \: \: \: \: \: - - - - (3)$
• $\sf{x + y = - 11} \: \: \: \: \: \: - - - - (1)$

• On adding Both Equation y term Cancel

${~~~~~:~~\implies \sf 2x+y-y = 21 - 11}$

${~~~~~:~~\implies \sf 2x\cancel{+y}\cancel{-y} = 21 - 11}$

${~~~~~:~~\implies \sf 2x = 21 - 11}$

${~~~~~:~~\implies \sf 2x = 10}$

${~~~~~:~~\implies \sf x = 5}$

Putting x= 5 in Equation 1

${~~~~~:~~\implies \sf x + y = - 11}$

${~~~~~:~~\implies \sf 5+ y = - 11}$

${~~~~~:~~\implies \sf y = - 11 - 5}$

${~~~~~:~~\implies \sf y = -16}$

• Hance x= 5 and y = -16 .

Verification

$\small\small\small\small\small \sf{ \: \: \: \pink{x + y = - 11} ~~~~~~~~~~~\: \: \: \: \: or\green{= \: \: \: \: \: xy = - 80}}$

$\small\small\small\small\small\small\small\sf{ \: \: \:\pink{ 5 +( - 16) = - 11}} \: \: \: \: \: or \: \: \: \: \:\green{ 5 \times ( - 16)= - 80}$

$\small\small\small\small\small\small\small{ \: \: \: \pink{ - 11 = - 11\: \: \: _{_{_{ verifed}}}} \: \: \: \: \: or \: \: \: \: \: \green{- 80= - 80 \: \: \: _{_{_{ verifed}}}}}$

$\begin{gathered}\begin{gathered}\\\\\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \bigstar \: \underline{\sf{\red{Happy~ Studying}}\bigstar}\\ {\boxed{\begin{array}{cc}\dashrightarrow \sf(a + b)^{2} = {a}^{2} + {b}^{2} + 2ab \\\dashrightarrow \sf(a - b)^{2} = {a}^{2} + {b}^{2} - 2ab \\\dashrightarrow \sf(a + b)(a - b) = {a}^{2} - {b}^{2} \\\dashrightarrow \sf(a + b) ^{3} = {a}^{3} + b^{3} + 3ab(a + b) \\ \dashrightarrow\sf(a - b) ^{3} = {a}^{3} - b^{3} - 3ab(a - b) \\ \dashrightarrow\sf a ^{3} + {b}^{3} = (a + b)(a ^{2} + {b}^{2} - ab) \\\dashrightarrow \sf a ^{3} - {b}^{3} = (a - b)(a ^{2} + {b}^{2} + ab \\\dashrightarrow \sf{a²+b²=(a+b)²-2ab}\\ \end{array}}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$