Factor the Following:
3x^2 + 10xy + 7y^2
3x^2 + 10xy + 7y^2
(x + y )(3x + 7y)
To factor the expression 3x^2 + 10xy + 7y^2, we need to find two binomials that when multiplied together will give us the given expression.
First, let's check if the expression can be factored further by looking at the coefficients of x^2, xy, and y^2.
The coefficient of x^2 is 3, which cannot be factored further since it is a prime number.
The coefficient of y^2 is 7, which also cannot be factored further since it is a prime number.
Now, let's find two binomials in the form (ax + by)(cx + dy) that when multiplied together, will give us the given expression.
The first terms of the binomials will be (3x)(x) = 3x^2.
The last terms of the binomials will be (7y)(y) = 7y^2.
To find the middle term, we need to find two numbers that multiply to give us 7y^2 and add up to 10xy. Since the coefficient of y^2 is 7, the only possible product for the numbers is 7y * y. And since there is only one combination of numbers that can be multiplied to give 7y^2, and that is 7y * y, we can factor like so:
(3x^2 + 7y^2) + (10xy)
=> (3x^2 + 7y^2) + (7xy + 3xy)
=> (3x^2 + 7xy) + (3xy + 7y^2)
=> x(3x + 7y) + y(3x + 7y)
Now, we can see that we have two common terms (3x + 7y). Therefore, we can factor out (3x + 7y) from both terms:
(3x + 7y)(x + y)
So, the factored form of the expression 3x^2 + 10xy + 7y^2 is (3x + 7y)(x + y).
To factor the quadratic expression 3x^2 + 10xy + 7y^2, we need to find two binomial expressions that, when multiplied together, result in the original quadratic expression. Here's how you can do it:
Step 1: Observe the quadratic expression: 3x^2 + 10xy + 7y^2
Step 2: Look for patterns or common factors. In this case, there are no common factors that can be factored out from all terms.
Step 3: We need to find two binomial expressions in the form of (ax + by)(cx + dy) that, when multiplied together, equal the original expression.
Step 4: Multiply the first terms of the binomials: a * c = 3x * 3x = 9x^2
Step 5: Multiply the last terms of the binomials: b * d = 7y * y = 7y^2
Step 6: Multiply the outer terms of the binomials: a * d = 3x * y = 3xy
Step 7: Multiply the inner terms of the binomials: b * c = 7y * 3x = 21xy
Step 8: Combine the middle terms: 3xy + 21xy = 24xy
Step 9: We now have: 9x^2 + 24xy + 7y^2
Step 10: The original expression matches the expanded form: 9x^2 + 24xy + 7y^2
Therefore, the factored form of 3x^2 + 10xy + 7y^2 is: (3x + y)(x + 7y)