Question

How do you determine the coefficients a and b such that $p\left( x \right) = x^2 + ax + b$ satisfies $p\left( 1 \right) = -7$ and $p’\left( 1 \right) = 4$ ?

Answer 1

Since you have p(1) = -7 in the question, you need to substitute -1 in p(x). After substituting, you get the equation in the terms of a and b. Next, you need to differentiate the p(x) in order to get the p’(x). Then again you use the initial condition given in the question. Now, you get the equation in terms of a . After you get the value of a, you need to substitute in the previous equation, to get the value of b.

Complete step by step solution:
Here is the complete step by step solution.
The first step we need to do is to substitute -1 in p(x), since you have p(1) = -7 in the question. After substituting, you get the equation in the terms of a and b. Therefore, we get
$\Rightarrow p\left( x \right) = x^2 + ax + b$
$\Rightarrow p\left( 1 \right) = 1+ a + b$
$\Rightarrow -7 = 1+ a + b$
$\Rightarrow a + b = -8$ --- (1)
Now, we need to differentiate the p(x) in order to get the p’(x). Then again we use the initial condition given in the question. Then we get the equation in terms of a .
$\Rightarrow p’\left( x \right) = 2x+ a$
$\Rightarrow p’\left( 1 \right) = 2+ a = 4$
$\Rightarrow a = 2$
Now we need substitute the value of a in the equation 1, to get the value of b
$\Rightarrow a+b=-8$
$\Rightarrow 2+b=-8$
$\Rightarrow b=-10$

Therefore, we get the final answer of the question, as a = 2 and b = -10.

Note: When you get these types of questions, you need to use the initial conditions given in the question, and you need to substitute them in the equation to get the answer. You need to be careful while doing the substitutions, or else you get the wrong answer.