Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x2 – 2x and g(f(x)) = 4x2 + 6x + 1, then the value of f(2) + g(2) is ____________ .
Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x2 – 2x and g(f(x)) = 4x2 + 6x + 1, then the value of f(2) + g(2) is ____________ .
Correct Answer: 18
Solution and Explanation
The correct answer is 18
f(g(x)) = 8x2 – 2x
g(f(x)) = 4x2 + 6x + 1
let f(x) = cx2 + dx + e
g(x) = ax + b
f(g(x)) = c(ax + b)2 + d(ax + b) + e ≡ 8x2 – 2x
g(f(x)) = a(cx2 + dx + e) + b ≡ 4x2 + 6x + 1
Therefore ac = 4 ad = 6 ae + b = 1
a2c = 8 2abc + ad = –2 cb2 + bd + e = 0
By solving, we get
a = 2 , b = –1, c = 2 , d = 3, e = 1
∴ f(x) = 2x2 + 3x +1
g(x) = 2x – 1
f(2) + g(2) = 2(2)2 + 3(2) + 1 + 2(2) – 1
= 18
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Concepts Used:
Quadratic Equations
A polynomial that has two roots or is of degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.
Consider the following equation ax²+bx+c=0, where a≠0 and a, b, and c are real coefficients.
The solution of a quadratic equation can be found using the formula, x=((-b±√(b²-4ac))/2a)
Two important points to keep in mind are:
- A polynomial equation has at least one root.
- A polynomial equation of degree ‘n’ has ‘n’ roots.
Read More: Nature of Roots of Quadratic Equation
There are basically four methods of solving quadratic equations. They are:
- Factoring
- Completing the square
- Using Quadratic Formula
- Taking the square root