Solve Using the Square Root Property x^2+(14-x)^2=106

Math
x2+(14-x)2=106
Move 106 to the left side of the equation by subtracting it from both sides.
x2+(14-x)2-106=0
Factor the left side of the equation.
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Rewrite (14-x)2 as (14-x)(14-x).
x2+(14-x)(14-x)-106=0
Expand (14-x)(14-x) using the FOIL Method.
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Apply the distributive property.
x2+14(14-x)-x(14-x)-106=0
Apply the distributive property.
x2+14⋅14+14(-x)-x(14-x)-106=0
Apply the distributive property.
x2+14⋅14+14(-x)-x⋅14-x(-x)-106=0
x2+14⋅14+14(-x)-x⋅14-x(-x)-106=0
Simplify and combine like terms.
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Simplify each term.
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Multiply 14 by 14.
x2+196+14(-x)-x⋅14-x(-x)-106=0
Multiply -1 by 14.
x2+196-14x-x⋅14-x(-x)-106=0
Multiply 14 by -1.
x2+196-14x-14x-x(-x)-106=0
Multiply x by x.
x2+196-14x-14x-1⋅(-1×2)-106=0
Multiply -1 by -1.
x2+196-14x-14x+1×2-106=0
Multiply x2 by 1.
x2+196-14x-14x+x2-106=0
x2+196-14x-14x+x2-106=0
Subtract 14x from -14x.
x2+196-28x+x2-106=0
x2+196-28x+x2-106=0
Add x2 and x2.
2×2+196-28x-106=0
Subtract 106 from 196.
2×2-28x+90=0
Rewrite 2×2-28x+90 in a factored form.
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Factor 2 out of 2×2-28x+90.
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Factor 2 out of 2×2.
2(x2)-28x+90=0
Factor 2 out of -28x.
2(x2)+2(-14x)+90=0
Factor 2 out of 90.
2×2+2(-14x)+2⋅45=0
Factor 2 out of 2×2+2(-14x).
2(x2-14x)+2⋅45=0
Factor 2 out of 2(x2-14x)+2⋅45.
2(x2-14x+45)=0
2(x2-14x+45)=0
Factor.
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Factor x2-14x+45 using the AC method.
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Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is 45 and whose sum is -14.
-9,-5
Write the factored form using these integers.
2((x-9)(x-5))=0
2((x-9)(x-5))=0
Remove unnecessary parentheses.
2(x-9)(x-5)=0
2(x-9)(x-5)=0
2(x-9)(x-5)=0
2(x-9)(x-5)=0
Divide each term by 2 and simplify.
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Divide each term in 2(x-9)(x-5)=0 by 2.
2(x-9)(x-5)2=02
Simplify 2(x-9)(x-5)2.
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Cancel the common factor of 2.
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Cancel the common factor.
2(x-9)(x-5)2=02
Divide (x-9)(x-5) by 1.
(x-9)(x-5)=02
(x-9)(x-5)=02
Expand (x-9)(x-5) using the FOIL Method.
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Apply the distributive property.
x(x-5)-9(x-5)=02
Apply the distributive property.
x⋅x+x⋅-5-9(x-5)=02
Apply the distributive property.
x⋅x+x⋅-5-9x-9⋅-5=02
x⋅x+x⋅-5-9x-9⋅-5=02
Simplify and combine like terms.
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Simplify each term.
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Multiply x by x.
x2+x⋅-5-9x-9⋅-5=02
Move -5 to the left of x.
x2-5⋅x-9x-9⋅-5=02
Multiply -9 by -5.
x2-5x-9x+45=02
x2-5x-9x+45=02
Subtract 9x from -5x.
x2-14x+45=02
x2-14x+45=02
x2-14x+45=02
Divide 0 by 2.
x2-14x+45=0
x2-14x+45=0
Factor x2-14x+45 using the AC method.
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Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is 45 and whose sum is -14.
-9,-5
Write the factored form using these integers.
(x-9)(x-5)=0
(x-9)(x-5)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
x-9=0
x-5=0
Set the first factor equal to 0 and solve.
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Set the first factor equal to 0.
x-9=0
Add 9 to both sides of the equation.
x=9
x=9
Set the next factor equal to 0 and solve.
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Set the next factor equal to 0.
x-5=0
Add 5 to both sides of the equation.
x=5
x=5
The final solution is all the values that make (x-9)(x-5)=0 true.
x=9,5
Solve Using the Square Root Property x^2+(14-x)^2=106

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