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Advanced Math / Nonlinear functions Difficulty: Hard

The area of a triangle is equal to $x^{2}$ square centimeters. The length of the base of the triangle is $2 x plus 22$ centimeters, and the height of the triangle is $x minus 10$ centimeters. What is the value of $x$ ?

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Explanation

The correct answer is $110$ . The area of a triangle is equal to one half of the product of the length of the base of the triangle and the height of the triangle. It's given that the length of the base of the triangle is $2 x plus 22$ centimeters and the height of the triangle is $x minus 10$ centimeters; therefore, its area is $\frac{1}{2} (2 x + 22) (x - 10)$ square centimeters. It's also given that the area of the triangle is equal to $x^{2}$ square centimeters. Therefore, it follows that $\frac{1}{2} (2 x + 22) (x - 10) = x^{2}$ . This equation can be rewritten as $(x + 11) (x - 10) = x^{2}$ , or $x^{2} + x - 110 = x^{2}$ . Subtracting $x^{2}$ from both sides of this equation yields $x - 110 = 0$ . Adding $110$ to both sides of this equation yields $x equals 110$ . Therefore, the value of $x$ is $110$ .