Factorize the following:
2x^2 + 3x - 7
This will not factor into rational roots.
Use the quadratic formula
To factorize the quadratic expression 2x^2 + 3x - 7, we need to find two binomial expressions that multiply together to give the original expression.
Step 1: Multiply the coefficient of x^2 (2) by the constant term (-7). In this case, (-7)(2) equals -14.
Step 2: Find two numbers that multiply together to give -14 and add up to the coefficient of x (3). In this case, the numbers are 7 and -2, since 7 * (-2) = -14 and 7 + (-2) = 5.
Step 3: Rewrite the middle term (3x) using the two numbers obtained in step 2. So, we can rewrite 3x as 7x - 2x.
Now, we can factorize the expression:
2x^2 + 7x - 2x - 7
Group the terms:
(2x^2 + 7x) + (-2x - 7)
Factor out the common factors from each group:
x(2x + 7) - 1(2x + 7)
Now, notice that (2x + 7) is a common factor of both terms. We can factor it out:
(2x + 7)(x - 1)
Therefore, the factored form of the expression 2x^2 + 3x - 7 is (2x + 7)(x - 1).