How do you factor 12x^2+8x-15? Thanks in advance.
Look for combinations of 4 and 3, or 6 and 2, for the x term of one monomial factor, and 5 and 3 for the constant term, and see what works
Example:
(4x -3)(3x +5) gives 12 x^2 +11 x -15
(6x -5)(2x +3) gives ?
It's a trial and error process of picking suitable factors that yield the right coefficients for the x^2 and constant terms, and seeing what the middle term coefficient of x turns out to be.
If that doesn't work, you can always go back to the equation
x = [-b +/2 sqrt(b^2-4ac)]/2a and taking the opposite sign of the roots. If the roots are m and n, the equation
ax^2 + bx + c = 0 factors to a(x-m)(x-n) = 0.
Couldn't you also have (12x-__)(1x+___) or (__-15)(___+3), etc? Isn't there an easier way then this guess and check process???
To factor the expression 12x^2 + 8x - 15, we can use a method called "factoring by grouping." Here's how you can do it:
Step 1: Multiply the coefficient of the first term and the constant term: 12 * -15 = -180.
Step 2: Find two numbers whose product is equal to -180 and whose sum is equal to the coefficient of the middle term, which is 8. In this case, the numbers are 20 and -9 because 20 * -9 = -180 and 20 + (-9) = 11.
Step 3: Rewrite the middle term, 8x, as the sum of the two numbers found in step 2. So, instead of 8x, we will write it as 20x - 9x.
Step 4: Group the terms into two pairs:
(12x^2 + 20x) + (-9x - 15)
Step 5: Factor out the greatest common factor from each pair:
4x(3x + 5) - 3(3x + 5)
Step 6: Notice that we have a common binomial factor, (3x + 5), in both terms. Factor it out:
(3x + 5)(4x - 3)
So, the factored form of 12x^2 + 8x - 15 is (3x + 5)(4x - 3).