Find the lengths of the sides of a triangle, if the side lengths are represented as x, x + 7, and 2x + 1.

**Use the Pythagorean Theorem**

a² + b² = c²

a² + b² = c²

a and b are the legs

c is the hypotenuse {the longest side}

x and x + 7 are the legs

2x + 1 is the hypotenuse

x² + (x + 7)² = (2x + 1)² {substituted into Pythagorean Theorem}

x² + x² + 14x + 49 = 4x² + 4x + 1 {squared the binomials}

2x² + 14x + 49 = 4x² + 4x + 1 {combined like terms}

2x² - 10x - 48 = 0 {subtracted 2x², 14x, and 49 from each side}

2(x² - 5x - 24) = 0 {factored 2 out of left side}

x² - 5x - 24 = 0 {divided each side by 2}

(x - 8)(x + 3) = 0 {factored into two binomials}

x - 8 = 0 or x + 3 = 0 {set each factor equal to zero}

x = 8 or x = -3 {solved each equation for x}

the length of a side cannot be negative

x = 8

x + 7 = 15

2x + 1 = 17

**8, 15, and 17**are the lengths of the sides

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