1. **State the problem:** We are given two expressions, $9k + 7$ and $2k^2 + 3$, which represent consecutive integers. We know $9k + 7$ is the smaller integer. 2. **Write the relationship for consecutive integers:** If $9k + 7$ is the smaller integer, then the next consecutive integer is $9k + 7 + 1 = 9k + 8$. 3. **Set up the equation:** Since $2k^2 + 3$ is the next consecutive integer, it must equal $9k + 8$. $$2k^2 + 3 = 9k + 8$$ 4. **Rearrange the equation to standard quadratic form:** $$2k^2 + 3 - 9k - 8 = 0$$ $$2k^2 - 9k - 5 = 0$$ 5. **Solve the quadratic equation using the quadratic formula:** The quadratic formula is: $$k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=-9$, and $c=-5$. Calculate the discriminant: $$b^2 - 4ac = (-9)^2 - 4 \times 2 \times (-5) = 81 + 40 = 121$$ Calculate $k$: $$k = \frac{-(-9) \pm \sqrt{121}}{2 \times 2} = \frac{9 \pm 11}{4}$$ 6. **Find the two possible values of $k$:** - For $k = \frac{9 + 11}{4} = \frac{20}{4} = 5$ - For $k = \frac{9 - 11}{4} = \frac{-2}{4} = -\frac{1}{2}$ 7. **Check which $k$ gives integer values for the consecutive integers:** - For $k=5$: $$9k + 7 = 9 \times 5 + 7 = 45 + 7 = 52$$ $$2k^2 + 3 = 2 \times 25 + 3 = 50 + 3 = 53$$ These are consecutive integers 52 and 53. - For $k = -\frac{1}{2}$: $$9k + 7 = 9 \times (-\frac{1}{2}) + 7 = -4.5 + 7 = 2.5$$ $$2k^2 + 3 = 2 \times \left(-\frac{1}{2}\right)^2 + 3 = 2 \times \frac{1}{4} + 3 = 0.5 + 3 = 3.5$$ These are not integers. 8. **Conclusion:** The valid value of $k$ is 5, and the next consecutive integer is: $$9k + 8 = 9 \times 5 + 8 = 45 + 8 = 53$$ **Final answer:** The next consecutive integer is 53.