Factor completely 3x2 + x + 7.
Solution:
After visual inspection of the given quadratic equation i.e. 3x2 + x + 7 one can infer that it cannot be factorized.
The next step therefore will be to identify the nature of roots. This can be done by using the formula to find the root of the quadratic equation of the form ax2 + bx + c. The quadratic formula is given as:
x = -b ± √(b2 - 4ac)/2a --- (1)
Since b2 - 4ac determines the nature of the roots of the quadratic equation, it is called the “Discriminant” of the quadratic equation. We denote it by the symbol △.
The values of a, b and c for the given quadratic equation 3x2 + x + 7 are as follows:
a = 3; b = 1; c = 7
The value of (b2 - 4ac) can be calculated as:
△ = 12 - 4(3)(7) = -83
We also know that,
| When △ < 0 |
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| When △ = 0 |
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| When △ > 0 | The roots are real and unequal |
x = -1/6 ± √-83/6. Since △ is a negative number its square root will be an imaginary number. Hence the roots of the quadratic equation will be complex and unequal as shown below:
⇒ x is either -1/6 + i√83/6 or -1/6 - i√83/6, where i = √-1
In addition, if a, b, and c are rational, △ < 0, and △ ≠ a perfect square, the roots for the equation are p + iq, p - iq. Also, p is rational and q is irrational.
In our solution p = -1/6 and q is √83/6 which is irrational.
Hence it is verified that the given equation cannot be factorized for real and rational solutions.
Factor completely 3x2 + x + 7.
Summary:
The factors for the 3x2 + x + 7 are -1/6 + i√83/6 or -1/6 - i√83/6 where i = √-1.
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