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How do you simplify the product \[(2x-5)(x+4)\] and write it in a standard form?

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Last updated date: 22nd Jun 2024
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Answer
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Hint: We can simplify the given expression by multiplying each component present in the first parenthesis to components present in the second parenthesis.. The obtained expression will then have to be written in the standard form which is \[a{{x}^{2}}+bx+c\], and this will be the required form of the expression.

Complete step-by-step solution:
According to the question given we have to simplify, to begin with we will start with opening the parenthesis and multiplying each component with another.
So we have,
\[(2x-5)(x+4)\]
\[\Rightarrow 2x(x+4)-5(x+4)\]
So we have opened the parenthesis of \[(2x-5)\], so that each component gets multiplied to the component in the adjoining parenthesis.
Now, we will multiply \[2x\] with \[(x+4)\] as well as \[5\] with \[(x+4)\], we get
\[\Rightarrow (2{{x}^{2}}+8x)-(5x+20)\]
Multiplication produced an entity with the power of x as 2, so the obtained equation is a quadratic equation.
Now, we will open up the brackets and the negative sign before \[(5x+20)\] will cause a sign reversal and similar terms getting reduced.
We get,
\[\Rightarrow 2{{x}^{2}}+8x-5x-20\]
Subtracting \[8x-5x\], we get the equation as
\[\Rightarrow 2{{x}^{2}}+3x-20\]
The standard form of a quadratic equation is \[a{{x}^{2}}+bx+c\], where ‘a’, ‘b’ and ‘c’ is a constant and ‘a’ is not equal to zero.
So the simplified expression in the standard form:
\[2{{x}^{2}}+3x-20\]

Note: we can also simplify in the way as follows:
\[(2x-5)(x+4)\]
We can convert this in the form, \[(x+a)(x+b)\], then we can use the formula \[(x+a)(x+b)={{x}^{2}}+(a+b)x+ab\]
Taking 2 common from \[(2x-5)\], we get
\[\Rightarrow 2\left( x-\dfrac{5}{2} \right)(x+4)\]
\[\Rightarrow 2\left( x+\left( -\dfrac{5}{2} \right) \right)(x+4)\]
Applying the formula \[(x+a)(x+b)={{x}^{2}}+(a+b)x+ab\], we have
\[\Rightarrow 2\left( {{x}^{2}}+\left( \left( -\dfrac{5}{2} \right)+4 \right)x+\left( -\dfrac{5}{2} \right)\times 4 \right)\]
\[\Rightarrow 2\left( {{x}^{2}}+\left( \dfrac{-5+8}{2} \right)x+\left( -\dfrac{5\times 4}{2} \right) \right)\]
\[\Rightarrow 2\left( {{x}^{2}}+\left( \dfrac{3}{2} \right)x+\left( -\dfrac{20}{2} \right) \right)\]
Multiplying 2 with the entire equation we obtained, we then have
\[\Rightarrow 2{{x}^{2}}+3x-20\]
Therefore, the required simplified form of the expression in the standard form is \[2{{x}^{2}}+3x-20\].