If (x - 3) is a factor of \(2x^{2} - 2x + p\), find the value of constant p.
Question 1 Report
If (x - 3) is a factor of \(2x^{2} - 2x + p\), find the value of constant p.
Answer Details
If (x-3) is a factor of the quadratic expression, then the expression can be factored into the form:
$$(x - 3)(ax + b)$$
where a and b are constants. Multiplying out the brackets gives:
$$2x^{2} - 2x + p = (x - 3)(ax + b) = ax^{2} + (b - 3a)x - 3b$$
Since the coefficients of the quadratic expression are equal to those of the factored expression, we can equate the corresponding coefficients to get a system of equations:
$$a = 2$$
$$b - 3a = -2$$
$$-3b = p$$
Solving these equations simultaneously, we get:
$$a = 2, b = 3a - 2 = 4, p = -3b = -12$$
Therefore, the constant p is -12.
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