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# If x is real, the maximum value of expression (x^{2 }+ 14x + 9) / (x^{2 }+ 2x + 3) will be what?

Algebra is the branch of maths dealing with expressions and variables. It has many applications. In this article, we will calculate the maximum value of the given expression.

## Answer: If x is real, the maximum value of expression (x^{2 }+ 14x + 9) / (x^{2 }+ 2x + 3) will be 4.

Let's understand the solution in detail.

**Explanation:**

Let y = (x^{2 }+ 14x + 9) / (x^{2 }+ 2x + 3)

⇒ y (x^{2 }+ 2x + 3) = (x^{2 }+ 14x + 9)

⇒ x^{2}y + 2xy + 3y = x^{2 }+ 14x + 9

⇒ (1 – y)x^{2} + 2x (7 – y) + 3(3 – y) = 0

Now, we see that the above equation is quadratic with x as the variable.

The equation is in the form ax^{2} + bx + c = 0

Now, we know that for real value of x, b^{2} - 4ac ≥ 0.

⇒ [2(7 – y)]^{2} – 4(1 – y) × 3(3 – y) ≥ 0

⇒ y^{2} + y - 20 ≤ 0

⇒ (y - 4) (y + 5) ≤ 0

Hence, we see that the solution of y lies in the range [-5, 4].

### Therefore, the maximum value of expression (x^{2 }+ 14x + 9) / (x^{2 }+ 2x + 3) will be 4 for real x.

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