The given equation is |9x^2 + 12x – 7| = k. Step 1: Let's first solve the quadratic equation 9x^2 + 12x – 7 = 0. Step 2: The roots of the equation can be found using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a. Here, a = 9, b = 12, and c = -7. Step 3: Substituting these values into the formula, we get x = [-12 ± sqrt((12)^2 - 4*9*(-7))] / 2*9. Step 4: Simplifying this, we get x = [-12 ± sqrt(144 + 252)] / 18 = [-12 ± sqrt(396)] / 18. Step 5: So, the roots of the equation are x = [-12 + sqrt(396)] / 18 and x = [-12 - sqrt(396)] / 18. Step 6: Now, the given equation |9x^2 + 12x – 7| = k will have four distinct real roots if and only if the quadratic equation 9x^2 + 12x – 7 = k and 9x^2 + 12x – 7 = -k both have real roots. Step 7: For a quadratic equation ax^2 + bx + c = 0 to have real roots, the discriminant b^2 - 4ac must be greater than or equal to 0. Step 8: So, for the equation 9x^2 + 12x – 7 = k to have real roots, we must have (12)^2 - 4*9*(k+7) = 0, which simplifies to 144 - 36k = 0, or k = 0, which simplifies to 144 + 36k = 0, or k = -4. Step 10: Combining these results, we find that the integer values of k for which the given equation has exactly four distinct real roots are k = -4, -3, -2, -1, 0, 1, 2, 3, 4. So, there are 9 integer values that k can take.

Question

The given equation is |9x^2 + 12x – 7| = k.

Step 1: Let's first solve the quadratic equation 9x^2 + 12x – 7 = 0.

Step 2: The roots of the equation can be found using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a. Here, a = 9, b = 12, and c = -7.

Step 3: Substituting these values into the formula, we get x = [-12 ± sqrt((12)^2 - 4*9*(-7))] / 2*9.

Step 4: Simplifying this, we get x = [-12 ± sqrt(144 + 252)] / 18 = [-12 ± sqrt(396)] / 18.

Step 5: So, the roots of the equation are x = [-12 + sqrt(396)] / 18 and x = [-12 - sqrt(396)] / 18.

Step 6: Now, the given equation |9x^2 + 12x – 7| = k will have four distinct real roots if and only if the quadratic equation 9x^2 + 12x – 7 = k and 9x^2 + 12x – 7 = -k both have real roots.

Step 7: For a quadratic equation ax^2 + bx + c = 0 to have real roots, the discriminant b^2 - 4ac must be greater than or equal to 0.

Step 8: So, for the equation 9x^2 + 12x – 7 = k to have real roots, we must have (12)^2 - 4*9*(k+7) >= 0, which simplifies to 144 - 36k >= 0, or k <= 4.

Step 9: Similarly, for the equation 9x^2 + 12x – 7 = -k to have real roots, we must have (12)^2 - 4*9*(-k+7) >= 0, which simplifies to 144 + 36k >= 0, or k >= -4.

Step 10: Combining these results, we find that the integer values of k for which the given equation has exactly four distinct real roots are k = -4, -3, -2, -1, 0, 1, 2, 3, 4.

So, there are 9 integer values that k can take.

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