Given Gx=x4+2x3-7x2-8x+12 . When Gx Is Divided By - Gauthmath

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Question

4/7 (4x+7)(x+sqrt(4k+7))(x-sqrt(4k+7))=0 In the given equation, k is a positive constant. The product of the solutions to the equation is 77. What is the value of k?SHOW LESS0

Solution

The answer is 37/4

Step 1: Identify the roots of the equation The given equation is $\frac {4}{7}(4x+7)(x+\sqrt {4k+7})(x-\sqrt {4k+7})=0$. The roots are the values of ( x ) that make the equation equal to zero. From the factor ( (4x + 7) ), we get ( 4x + 7 = 0 ), so ( x_1 = -\frac{7}{4} ). From the factor ( (x + \sqrt{4k + 7}) ), we get ( x_2 = -\sqrt{4k + 7} ). From the factor ( (x - \sqrt{4k + 7}) ), we get ( x_3 = \sqrt{4k + 7} ).

Step 2: Calculate the product of the roots The product of the roots is given as 77. Therefore, ( x_1 \cdot x_2 \cdot x_3 = \left(-\frac{7}{4}\right) \cdot \left(-\sqrt{4k+7}\right) \cdot \left(\sqrt{4k+7}\right) = 77 ) ( \left(-\frac{7}{4}\right) \cdot \left(- (4k+7)\right) = 77 ) ( \frac{7}{4} (4k+7) = 77 )

Step 3: Solve for k Multiply both sides by ( \frac{4}{7} ): ( 4k + 7 = 77 \cdot \frac{4}{7} ) ( 4k + 7 = 11 \cdot 4 ) ( 4k + 7 = 44 ) Subtract 7 from both sides: ( 4k = 44 - 7 ) ( 4k = 37 ) Divide by 4: ( k = \frac{37}{4} )

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