Exercise 17 Page 772

We want to find the value of x.

Let's recall that if two secants intersect in the exterior of a circle, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment. Using this information we can write an equation for our circle. ( 8+x+4)* 8=(2x+x)* xLet's solve this equation for x.

(8+x+4)* 8= (2x+x)* x

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Solve for x

AddTerms

Add terms

(12+x)* 8=3x * x

Multiply

Multiply

(12+x)*8=3x^2

Distr

Distribute 8

96+8x=3x^2

SubEqn

LHS-3x^2=RHS-3x^2

- 3x^2+96+8x=0

CommutativePropAdd

Commutative Property of Addition

- 3x^2+8x+96=0

Note that we end with a quadratic equation. To solve this equation we can use the Quadratic Formula. ax^2+ bx+ c=0 ⇔ x=- b± sqrt(b^2-4 a c)/2 a We first need to identify the values of a, b, and c. - 3x^2+8x+96=0 ⇔ - 3x^2+ 8x+ 96=0 We see that a= - 3, b= 8, and c= 96. Let's substitute these values into the Quadratic Formula.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=- 8±sqrt(8^2-4( - 3)( 96))/2( - 3)

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Solve for x and Simplify

CalcPow

Calculate power

x=- 8±sqrt(64-4(-3)(96))/2(- 3)

Multiply

Multiply

x=- 8±sqrt(64+1152)/- 6

AddTerms

Add terms

x=- 8±sqrt(1216)/- 6

SplitIntoFactors

Split into factors

x=- 8±sqrt(64* 19)/- 6

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

x=- 8 ± sqrt(64)* sqrt(19)/- 6

CalcRoot

Calculate root

x=- 8 ± 8 sqrt(19)/- 6

FactorOut

Factor out 2

x=2(- 4± 4sqrt(19))/2(- 3)

CancelCommonFac

Cancel out common factors

x=- 4± 4sqrt(19)/- 3

DivNegNeg

- a/- b=a/b

x=4± 4sqrt(19)/3

Using the Quadratic Formula, we found that the solutions of the given equation are x= 4± 4sqrt(19)3. Thus, the solutions are x_1= 4 + 4sqrt(19)3≈ 7.1 and x_2= 4 - 4sqrt(19)3≈ - 4.5. Since the length of a segment is always a positive number, the only correct solution is x_1≈ 7.1.