Question

What is the value of $$a{x^2} + bx + c$$ at $$x = \dfrac{{ - b}}{a}$$?
1). $$a$$
2). $${b^2} - 4ac$$
3). $$c$$
4). $$0$$

Answer 1

Hint: Here in this question, we have to find the value of a given algebraic or quadratic expression by using the given values of variables. To find this we have to substitute the x values in the given quadratic expression and by further simplification using a basic arithmetic operation to get the required solution.

Complete step-by-step solution:
Algebraic expression is a mathematical term that consists of variables and constants along with mathematical operators (subtraction, addition, multiplication, etc).
A quadratic expression is an algebraic polynomial having variable degree 2.
consider the given algebraic expression:
$$ \Rightarrow \,f\left( x \right) = \,a{x^2} + bx + c$$ -----(1)
An expression with three terms is known as trinomial.
Here ‘x’ be the variables whose value is unknown, it can vary depending upon situations and a, b, c are coefficients of the variables it will remain unchanged.
From the question
We know that the value of variable $$x = \dfrac{{ - b}}{a}$$. Substitute the value of “x” in the given equation:
$$ \Rightarrow f\left( {\dfrac{{ - b}}{a}} \right) = \,\,a{\left( {\dfrac{{ - b}}{a}} \right)^2} + b\left( {\dfrac{{ - b}}{a}} \right) + c$$
By the properties of exponent $${\left( {\dfrac{a}{b}} \right)^n} = \dfrac{{{a^n}}}{{{b^n}}}$$, then we have
$$ \Rightarrow \,\,f\left( {\dfrac{{ - b}}{a}} \right) = a\left( {\dfrac{{{{\left( { - b} \right)}^2}}}{{{a^2}}}} \right) + b\left( {\dfrac{{ - b}}{a}} \right) + c$$
$$ \Rightarrow f\left( {\dfrac{{ - b}}{a}} \right) = \,\,a\left( {\dfrac{{{b^2}}}{{{a^2}}}} \right) + b\left( {\dfrac{{ - b}}{a}} \right) + c$$
On multiplication, we have
$$ \Rightarrow \,\,\,f\left( {\dfrac{b}{a}} \right) = \,\,\dfrac{{{b^2}}}{a} - \dfrac{{{b^2}}}{a} + c$$
On simplification, we get
$$\therefore \,\,f\left( {\dfrac{{ - b}}{a}} \right) = c$$
Hence, the value of $$a{x^2} + bx + c$$ at $$x = \dfrac{{ - b}}{a}$$ is $$c$$.
Therefore, option (3) is the correct answer.

Note: The value of algebraic expression will vary as the value of a variable varies. A variable is a letter whose value is unknown, it can take any value depending upon the situation. If the variable's value is integer, then we should take care of the sign of the algebraic expression using a sign convention and simplify algebraic expression using mathematical operations.