Question

Solve for x using factorization method
${x}^{2} + x = (n + 1)(n + 2)$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given quadratic equation is

$\rm \: {x}^{2} + x - (n + 1)(n + 2) = 0$

can be rewritten as

$\rm \: {x}^{2} + \red{(2 - 1)}x - (n + 1)(n + 2) = 0$

can be further rewritten as

$\rm \: {x}^{2} + \red{(2 - 1 + n - n)}x - (n + 1)(n + 2) = 0$

can be re-arranged as

$\rm \: {x}^{2} + \red{[(n + 2) - (n + 1)]}x - (n + 1)(n + 2) = 0$

$\rm \: {x}^{2} + (n + 2)x - (n + 1)x - (n + 1)(n + 2) = 0$

$\rm \: x[x + (n + 2)] - (n + 1)[x + (n + 2)] = 0$

$\rm \: [x + (n + 2)] [x- (n + 1)] = 0$

$\rm \: x + n + 2 = 0 \: \: \: or \: \: \: x - (n + 1) = 0$

$\rm\implies \:x = - (n + 2) \: \: \: or \: \: \: x = n + 1$

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Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac

Answer 2

Given quadratic equation is

$\begin{gathered}\rm \: {x}^{2} + x - (n + 1)(n + 2) = 0 \\ \end{gathered}$

can be rewritten as

$\begin{gathered}\rm \: {x}^{2} + \red{(2 - 1)}x - (n + 1)(n + 2) = 0 \\ \end{gathered}$

can be further rewritten as

$\begin{gathered}\rm \: {x}^{2} + \red{(2 - 1 + n - n)}x - (n + 1)(n + 2) = 0 \\ \end{gathered}$

can be re-arranged as

$\begin{gathered}\rm \: {x}^{2} + \red{[(n + 2) - (n + 1)]}x - (n + 1)(n + 2) = 0 \\ \end{gathered} \\ \begin{gathered}\rm \: {x}^{2} + (n + 2)x - (n + 1)x - (n + 1)(n + 2) = 0 \\ \end{gathered} \\ \begin{gathered}\rm \: x[x + (n + 2)] - (n + 1)[x + (n + 2)] = 0 \\ \end{gathered} \\ \begin{gathered}\rm \: x + n + 2 = 0 \: \: \: or \: \: \: x - (n + 1) = 0 \\ \end{gathered} \\ \begin{gathered}\rm\implies \:x = - (n + 2) \: \: \: or \: \: \: x = n + 1 \\ \end{gathered}$

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Additional Information :-

• Nature of roots :-

• Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac