HOW TO FACTOR 20X^2+23X-21 STEP BY STEP
you have
20x^2 + 23x - 21
if p/q is a root, then p divides 21 and q divides 24. So, there will be lots of things to try.
since the 20 and 23 are sort of close to each other, try factors of 20 that are sort of close: 4 and 5
(4x+?)(5x-?) or (4x-?)(5x+?)
the ?'s will divide 21, but it's basically trial and error. A bit of effort will show that (4x+7)(5x-3) works
Usually with messy one like this, the good old quadratic formula is easiest
x = (-23 ± √(23^2 + 4*20*21))/40
x = (-23 ± √(2209))/40
x = (-23 ± 47)/40
x = 24/40 or -70/40
so,
(40x-24) = 0
(40x+70) = 0
pull out factors to get
(5x-3) = 0
(4x+7) = 0
Those are the factor
To factor the quadratic expression 20x^2 + 23x - 21 step-by-step, follow these steps:
Step 1: Determine if the expression can be factored using the AC method. The coefficient of the x^2 term is 20, and the constant term is -21. Multiply these two values: 20 * -21 = -420.
Step 2: Look for two numbers whose product is equal to -420 and whose sum is equal to the coefficient of the x term (23). After analyzing the factors of -420, we find that the numbers 28 and -15 satisfy this condition.
Step 3: Rewrite the middle term (23x) using the two numbers found in Step 2. Replace the middle term of the expression 20x^2 + 23x - 21 with +28x - 15x.
20x^2 + 28x - 15x - 21
Step 4: Group the terms in pairs, using parentheses.
(20x^2 + 28x) + (-15x - 21)
Step 5: Factor out the greatest common factor from each pair.
4x(5x + 7) - 3(5x + 7)
Step 6: Notice that in both terms, 5x + 7 appears as a common factor. Factor this out.
(4x - 3)(5x + 7)
Therefore, the factored form of the expression 20x^2 + 23x - 21 is (4x - 3)(5x + 7).
To factor the expression 20x^2 + 23x - 21 step by step, follow these instructions:
Step 1: Look for two numbers whose product is equal to the product of the leading coefficient (20) and the constant term (-21), which in this case is -420. The sum of these two numbers should be equal to the coefficient of the middle term (23).
Step 2: Write down all the possible pairs of numbers that multiply to give -420. In this case, some possible pairs are: (-1, 420), (1, -420), (-2, 210), (2, -210), (-3, 140), (3, -140), (-4, 105), (4, -105), (-5, 84), (5, -84), (-6, 70), (6, -70), (-7, 60), (7, -60), (-10, 42), (10, -42), (-12, 35), (12, -35), (-14, 30), (14, -30), (-15, 28), (15, -28), (-20, 21), and (20, -21).
Step 3: Test each pair of numbers by adding them together to see if the sum matches the middle term's coefficient (23). In this example, the pair (4, -105) adds up to 4 + (-105) = -101, which is not equal to 23. So, we continue testing the other pairs.
Step 4: After testing all the pairs, you will find that the pair (7, -60) adds up to 7 + (-60) = -53, which is equal to 23. This means that 7 and -60 are the numbers we need.
Step 5: Rewrite the middle term (23x) using the two numbers you found in step 4: 23x = 7x - 60x.
Step 6: Split the original expression using the rewritten middle term: 20x^2 + 7x - 60x - 21.
Step 7: Factor by grouping. Now, we group the terms in pairs and factor out the greatest common factor from each pair. The pairs are: (20x^2 + 7x) and (-60x - 21).
Step 8: From the first group (20x^2 + 7x), we can factor out the greatest common factor, which is x, giving us: x(20x + 7).
Step 9: From the second group (-60x - 21), we can factor out the greatest common factor, which is -3, giving us: -3(20x + 7).
Step 10: Combine the factored terms from step 8 and step 9: (x - 3)(20x + 7).
So, the factored expression for 20x^2 + 23x - 21 is (x - 3)(20x + 7).