Answer :
To factorize the quadratic expression \(x^2 + 27x + 162\), we follow these steps:
1. Identify the quadratic expression: \(x^2 + 27x + 162\).
2. Find the roots of the quadratic equation: To find factors, we look for values of \(x\) that satisfy the equation \(x^2 + 27x + 162 = 0\).
We use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
where \(a = 1\), \(b = 27\), and \(c = 162\).
3. Calculate the discriminant:
[tex]\[ b^2 - 4ac = 27^2 - 4(1)(162) = 729 - 648 = 81 \][/tex]
The discriminant is positive \(81\), implying two real roots.
4. Find the roots:
[tex]\[ x = \frac{{-27 \pm \sqrt{81}}}{2} = \frac{{-27 \pm 9}}{2} \][/tex]
[tex]\[ x = \frac{{-27 + 9}}{2} = \frac{{-18}}{2} = -9 \][/tex]
[tex]\[ x = \frac{{-27 - 9}}{2} = \frac{{-36}}{2} = -18 \][/tex]
Therefore, the roots are \(x = -9\) and \(x = -18\).
5. Factorize using the roots: If \(x = -9\) and \(x = -18\), we can write the factorized form as:
[tex]\[ (x + 9)(x + 18) \][/tex]
6. Expand the factorized form to confirm:
[tex]\[ (x + 9)(x + 18) = x^2 + 18x + 9x + 162 = x^2 + 27x + 162 \][/tex]
The factorization is indeed correct.
7. Extract individual terms from the factorized form: The terms are:
- From \((x + 9)\): \(x\) and \(9\).
- From \((x + 18)\): \(x\) and \(18\).
Thus, the complete individual terms involved in the factorization are \(9\), \(x\), \(18\), and \(x\).
Given the options, the correct answer is:
[tex]\[ \boxed{9, 9x, 18x} \][/tex]
1. Identify the quadratic expression: \(x^2 + 27x + 162\).
2. Find the roots of the quadratic equation: To find factors, we look for values of \(x\) that satisfy the equation \(x^2 + 27x + 162 = 0\).
We use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
where \(a = 1\), \(b = 27\), and \(c = 162\).
3. Calculate the discriminant:
[tex]\[ b^2 - 4ac = 27^2 - 4(1)(162) = 729 - 648 = 81 \][/tex]
The discriminant is positive \(81\), implying two real roots.
4. Find the roots:
[tex]\[ x = \frac{{-27 \pm \sqrt{81}}}{2} = \frac{{-27 \pm 9}}{2} \][/tex]
[tex]\[ x = \frac{{-27 + 9}}{2} = \frac{{-18}}{2} = -9 \][/tex]
[tex]\[ x = \frac{{-27 - 9}}{2} = \frac{{-36}}{2} = -18 \][/tex]
Therefore, the roots are \(x = -9\) and \(x = -18\).
5. Factorize using the roots: If \(x = -9\) and \(x = -18\), we can write the factorized form as:
[tex]\[ (x + 9)(x + 18) \][/tex]
6. Expand the factorized form to confirm:
[tex]\[ (x + 9)(x + 18) = x^2 + 18x + 9x + 162 = x^2 + 27x + 162 \][/tex]
The factorization is indeed correct.
7. Extract individual terms from the factorized form: The terms are:
- From \((x + 9)\): \(x\) and \(9\).
- From \((x + 18)\): \(x\) and \(18\).
Thus, the complete individual terms involved in the factorization are \(9\), \(x\), \(18\), and \(x\).
Given the options, the correct answer is:
[tex]\[ \boxed{9, 9x, 18x} \][/tex]
