Question

How do you write \[y=-2(x-9)(x+7)\] in standard form?

Answer 1

Hint: We know that the standard form of an equation is different for different types of equations. For a quadratic equation \[a{{x}^{2}}+bx=-c\], the standard form of this quadratic equation is \[a{{x}^{2}}+bx+c=0\]. For a linear equation like \[by=ax+c\], the standard form is \[ax-by=-c\]. And for an equation of the form \[y=ax(x-b)+c\], standard form is \[y=a{{x}^{2}}-abx+c\].

Complete step by step answer:
According to the given question, we have to find the standard form of the given equation \[y=-2(x-9)(x+7)\]. The given equation is same as that of the equation \[y=a(x-b)(x-c)\] whose standard form is the sum of \[{{x}^{2}}\] term, x term and constant, which is \[y=a{{x}^{2}}-a(b+c)x+abc=p{{x}^{2}}+qx+r\]. This representation same as that of the equation of polynomial \[y={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+......+{{a}_{n}}{{x}^{n}}\].

Now let us simplify the given equation to get the standard form.
Given, \[y=-2(x-9)(x+7)\]
For multiplying \[(x-9)\] with \[(x+7)\], we have to multiply x with \[(x+7)\] and -9 with \[(x+7)\] and then adding the term \[-9(x+7)\] with the other term \[x(x+7)\]. That means, we can write \[y=-2(x-9)(x+7)\] as
\[\Rightarrow y=-2\left( x\left( x+7 \right)-9(x+7) \right)\] ---------(1)
On multiplying x with \[(x+7)\], we get \[{{x}^{2}}+7x\]. That is, \[x(x+7)={{x}^{2}}+7x\] ----(2)
On multiplying -9 with \[(x+7)\], we get \[(-9x-63)\]. That is \[-9(x+7)=(-9x-63)\] ---(3)

Now, by substituting both equations (2) and (3) in equation (1), we get
\[\Rightarrow y=-2\left( {{x}^{2}}+7x-9x-63 \right)\] ---------(4)
By adding \[7x\] and \[-9x\], we get \[-2x\]. That is \[7x-9x=-2x\] ------(5)

Now, by substituting equation (5) in equation (4), we get
\[\Rightarrow y=-2({{x}^{2}}-2x-63)\] --------(6)

Now, we have to multiply the obtained equation \[({{x}^{2}}-2x-63)\] with -2. Here, we get
\[\Rightarrow y=-2{{x}^{2}}+4x+126\]

\[\therefore y=-2{{x}^{2}}+4x+126\] is the required standard form equation.

Note: In order to solve such types of questions, one must be aware of the standard form of different equations. While solving such types of problems, general error occurs in the calculation part especially while multiplying two terms. We must concentrate on the power of x terms when we multiply with different or same powers of x terms.