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

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.

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.

Simplify each term.

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.

Factor 2 out of 2×2-28x+90.

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.

Factor x2-14x+45 using the AC method.

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 in 2(x-9)(x-5)=0 by 2.

2(x-9)(x-5)2=02

Simplify 2(x-9)(x-5)2.

Cancel the common factor of 2.

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.

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.

Simplify each term.

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

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.

x-9=0

Add 9 to both sides of the equation.

x=9

x=9

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