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How much is \[\left( {4{x^2} - 6x + 7} \right) + \left( { - 19{x^2} - 15x - 18} \right)\]?

Last updated date: 23rd Feb 2024
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IVSAT 2024
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Hint: Here, we will remove the brackets and gather all the like terms and add or subtract them in order to get only a single term with a particular variable. By solving this further using mathematical operation, we will be able to find the required sum of the given two quadratic equations. Hence, this will be the required answer.

Complete step-by-step answer:
In order to find the sum of the given two quadratic equations, first of all, we will remove the parentheses as they are unnecessary and do not impact the question.
Thus, we get,
\[\left( {4{x^2} - 6x + 7} \right) + \left( { - 19{x^2} - 15x - 18} \right) = 4{x^2} - 6x + 7 - 19{x^2} - 15x - 18\]
Now, we will gather the like terms and solve them further by adding or subtracting whatever the case is.
\[ \Rightarrow \left( {4{x^2} - 6x + 7} \right) + \left( { - 19{x^2} - 15x - 18} \right) = \left( {4{x^2} - 19{x^2}} \right) + \left( { - 6x - 15x} \right) + \left( {7 - 18} \right)\]
\[ \Rightarrow \left( {4{x^2} - 6x + 7} \right) + \left( { - 19{x^2} - 15x - 18} \right) = - 15{x^2} - 21x - 11\]

Therefore, the required value of \[\left( {4{x^2} - 6x + 7} \right) + \left( { - 19{x^2} - 15x - 18} \right)\] is \[ - 15{x^2} - 21x - 11\]
Hence, this is the required answer.

Additional information:
In this question, we removed the parentheses. Parentheses are a pair of curved marks or brackets that we put around words or numbers to indicate that they are additional, separate, or the sign between them shows that the equations in the parentheses are being added, subtracted, divided or multiplied. Thus, it plays a vital role in the separation of two or more equations.

Quadratic equations are the polynomial equations of a second degree. This means that they contain at least one variable with a power or exponent two. The solutions of the quadratic equations are the values of unknown variables \[x\], which satisfies the equation. There are two possible solutions or roots of the quadratic equations. The number of solutions of an equation is equal to the highest degree of the equation.