Question

How do you write a quadratic equation with x-intercepts: -\[ - 4\], \[ - 2\]; point: \[\left( { - 6,8} \right)\]?

Answer 1

Hint: Here in this question, we have to find the quadratic equation with the given x-intercepts. This can be solve by using a equation of intercept form \[y = a\left( {x - p} \right)\left( {x - q} \right)\] where \[\left( {x - p} \right)\] and \[\left( {x - q} \right)\] are the factors by the given data \[p = - 4\] and \[q = - 2\] using this on simplification we get the required solution.

Complete step-by-step answer:
A quadratic equation in x is an equation that can be written in the standard form \[a{x^2} + bx + c = 0\], where a, b and c are real numbers and \[a \ne 0\].
‘a’ represents the numerical coefficient of \[{x^2}\],
‘b’ represents the numerical coefficient of \[x\], and
‘c’ represents the constant numerical term.
Consider a equation of intercept form
\[y = a\left( {x - p} \right)\left( {x - q} \right)\]--------(1)
Given the x intercepts i.e., \[p = - 4\] and \[q = - 2\] on substituting to the equation (1), we get
\[ \Rightarrow y = a\left( {x - \left( { - 4} \right)} \right)\left( {x - \left( { - 2} \right)} \right)\]
\[ \Rightarrow y = a\left( {x + 4} \right)\left( {x + 2} \right)\]
To find the value of a, plug or substitute the point \[\left( { - 6,8} \right)\] where x=-6 and y=8 , then
\[ \Rightarrow 8 = a\left( { - 6 + 4} \right)\left( { - 6 + 2} \right)\]
On simplification, we get
\[ \Rightarrow 8 = a\left( { - 2} \right)\left( { - 4} \right)\]
\[ \Rightarrow 8 = 8a\]
To isolate the a, divide both side by 8, then
\[ \Rightarrow a = \dfrac{8}{8}\]
\[ \Rightarrow a = 1\]
Now, we have the values of a=1, p=-4, and q=-2 all these values in equation (1), we get the required quadratic equation, then
\[ \Rightarrow y = 1\left( {x - \left( { - 4} \right)} \right)\left( {x - \left( { - 2} \right)} \right)\]
\[ \Rightarrow y = \left( {x + 4} \right)\left( {x + 2} \right)\]
Multiply the both factors in RHS
\[ \Rightarrow y = {x^2} + 2x + 4x + 8\]
\[ \Rightarrow y = {x^2} + 6x + 8\]
Hence, the required quadratic equation is \[y = {x^2} + 6x + 8\].
So, the correct answer is “\[y = {x^2} + 6x + 8\]”.

Note: The equation of intercept form \[y = a\left( {x - p} \right)\left( {x - q} \right)\] where \[\left( {x - p} \right)\] and \[\left( {x - q} \right)\] are the factors. The p and q are the roots of the equation. The general form of the quadratic equation is represented as \[a{x^2} + bx + c = 0\]. To simplify we use the simple arithmetic operations and we should know about the tables of multiplication.