factor the following quadratic expressions, if possible.
k^2-12k+20
6x^2 + 17x - 14
(k - 10)(k - 2)
(2x + 7)(3x - 2)
To factor a quadratic expression, we need to find two binomial factors that, when multiplied together, give us the original quadratic expression.
Let's start with the first expression:
k^2 - 12k + 20
To factor this quadratic expression, we need to find two numbers whose product is equal to the product of the coefficient of the squared term (k^2) and the constant term (20) and whose sum is equal to the coefficient of the linear term (-12k).
The coefficient of the squared term is 1, and the constant term is 20. The product of these numbers is 1 * 20 = 20.
Now, we need to find two numbers whose product is 20 and whose sum is -12. After considering different possibilities, we find that -2 and -10 satisfy these conditions.
So, we can rewrite the original expression as follows:
k^2 - 12k + 20
= (k - 2)(k - 10)
Therefore, the factored form of the expression k^2 - 12k + 20 is (k - 2)(k - 10).
Now let's move on to the second expression:
6x^2 + 17x - 14
To factor this quadratic expression, we need to find two numbers whose product is equal to the product of the coefficient of the squared term (6) and the constant term (-14) and whose sum is equal to the coefficient of the linear term (17x).
The coefficient of the squared term is 6, and the constant term is -14. The product of these numbers is 6 * -14 = -84.
Now, we need to find two numbers whose product is -84 and whose sum is 17. After considering different possibilities, we find that 21 and -4 satisfy these conditions.
So, we can rewrite the original expression as follows:
6x^2 + 17x - 14
= (3x - 4)(2x + 7)
Therefore, the factored form of the expression 6x^2 + 17x - 14 is (3x - 4)(2x + 7).
In both cases, we have successfully factored the given quadratic expressions.