1. **Stating the problem:** We have a right triangle with legs of lengths $x$ and 10, and hypotenuse $2x + 21$. We need to find the value of $x$. 2. **Formula used:** By the Pythagorean theorem, for a right triangle with legs $a$ and $b$ and hypotenuse $c$, we have: $$a^2 + b^2 = c^2$$ 3. **Apply the formula:** Here, $a = x$, $b = 10$, and $c = 2x + 21$. So: $$x^2 + 10^2 = (2x + 21)^2$$ 4. **Simplify:** $$x^2 + 100 = (2x + 21)^2$$ $$x^2 + 100 = 4x^2 + 84x + 441$$ 5. **Bring all terms to one side:** $$x^2 + 100 - 4x^2 - 84x - 441 = 0$$ $$-3x^2 - 84x - 341 = 0$$ 6. **Multiply both sides by $-1$ to simplify:** $$3x^2 + 84x + 341 = 0$$ 7. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=84$, $c=341$. 8. **Calculate the discriminant:** $$\Delta = 84^2 - 4 \times 3 \times 341 = 7056 - 4092 = 2964$$ 9. **Calculate the square root:** $$\sqrt{2964} = \sqrt{4 \times 741} = 2\sqrt{741} \approx 54.43$$ 10. **Calculate the two possible values for $x$:** $$x = \frac{-84 \pm 54.43}{6}$$ 11. **First solution:** $$x = \frac{-84 + 54.43}{6} = \frac{-29.57}{6} \approx -4.93$$ 12. **Second solution:** $$x = \frac{-84 - 54.43}{6} = \frac{-138.43}{6} \approx -23.07$$ 13. **Interpretation:** Since $x$ represents a length, it must be positive. Both solutions are negative, so we must check the problem setup or if the hypotenuse expression is correct. 14. **Re-examining the problem:** The hypotenuse is $2x + 21$, which must be greater than both legs. For positive $x$, $2x + 21$ is always positive and greater than $x$ and 10. 15. **Check for possible error:** The quadratic equation has no positive roots, so no valid $x$ satisfies the given conditions. **Final answer:** No positive value of $x$ satisfies the triangle with given sides.