Question

What are the solutions to the equation \[{x^2} + 6x = 40\] ?

Answer 1

Hint: Here given an algebraic equation firstly we have to convert the given equation to the quadratic equation \[a{x^2} + bx + c = 0\] by shifting 40 from RHS to LHS. Later solve the quadratic equation by using the method of factorisation and find the factors and equate each factor to the zero to get the value of \[x\] .

Complete step-by-step answer:
Factorization means the process of creating list of factors otherwise in mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original.
The general form of an equation is \[a{x^2} + bx + c\] when the coefficient of \[{x^2}\] is unity. Every quadratic equation can be expressed as \[{x^2} + bx + c = \left( {x + d} \right)\left( {x + e} \right)\] . Here, b is the sum of d and e and c is the product of d and e.
Consider the given expression:
 \[ \Rightarrow {x^2} + 6x = 40\]
Subtract both side by 40, then
 \[ \Rightarrow {x^2} + 6x - 40 = 40 - 40\]
On simplification, we get
 \[ \Rightarrow {x^2} + 6x - 40 = 0\]
The above equation is similar to a quadratic equation \[a{x^2} + bx + c\] now, solved by the method of factorization.
Now, Break the middle term as the summation of two numbers such that its product is equal to -40. Calculated above such two numbers are -4 and 10.
 \[ \Rightarrow {x^2} - 4x + 10x - 40 = 0\]
Making pairs of terms in the above expression
 \[ \Rightarrow \left( {{x^2} - 4x} \right) + \left( {10x - 40} \right) = 0\]
Take out greatest common divisor GCD from the both pairs, then
 \[ \Rightarrow x\left( {x - 4} \right) + 10\left( {x - 4} \right) = 0\]
Take \[\left( {x - 4} \right)\] common
 \[ \Rightarrow \left( {x - 4} \right)\left( {x + 10} \right) = 0\]
Equate the each factor to zero, then
 \[ \Rightarrow \left( {x - 4} \right) = 0\] or \[\left( {x + 10} \right) = 0\]
 \[ \Rightarrow x - 4 = 0\] OR \[x + 10 = 0\]
 \[ \Rightarrow x = 4\] OR \[x = - 10\]
Hence, the required solution is \[x = 4\] or \[x = - 10\] .
So, the correct answer is “\[x = 4\] or \[x = - 10\]”.

Note: The equation is a quadratic equation. This problem can be solved by using the sum product rule. This defines as for the general quadratic equation \[a{x^2} + bx + c\] , the product of \[a{x^2}\] and \[c\] is equal to the sum of \[bx\] of the equation. Hence, we obtain the factors. The factors for the equation depends on degree of the equation and this type of quadratic equation also solve by using completing the square method and also by using a quadratic formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] .