Consider the following quadratic equation:
12x^2=−2x
Step 1 of 2 : Using the standard form ax2+bx+c=0
of the given quadratic equation, factor the left hand side of the equation into two linear factors
To factorize the left hand side of the equation, we need to write the quadratic expression in the form (px + q)(rx + s) where p, q, r, and s are constants.
Given equation:
12x^2 = -2x
To factorize, let's first simplify the equation by dividing both sides by the greatest common factor, which is 2:
6x^2 = -x
Now we can see that the equation already has only one term on both sides. To factorize, we need to write it as a product of two linear factors.
Let's try the two linear factors to be (ax + b)(cx + d):
(6x^2) = (-2x)
(ax + b)(cx + d) = acx^2 + (ad + bc)x + bd
We can see that for the coefficient of x^2, ac must be equal to 6.
Let's try a = 2 and c = 3:
(2x + b)(3x + d) = 6x^2 + (6d + 3b)x + bd
By comparing the coefficients, we can obtain:
6d + 3b = -1 (Coefficient of the x term)
bd = 0 (Constant term)
To satisfy the first equation, one possibility is to let b = 0 and d = -1/6. However, this would make bd = 0, not -1.
Let's try another possibility:
Let b = -1 and d = 1/6:
(2x - 1)(3x + 1/6) = 6x^2 + (3/6 - 2/6)x - 1/6
Simplifying the equation:
(2x - 1)(3x + 1/6) = 6x^2 + (1/6)x - 1/6
Now we can rewrite the quadratic equation in the factored form:
(2x - 1)(3x + 1/6) = 0
So the factored form of the left hand side of the equation is (2x - 1)(3x + 1/6).