what is the expression in factored form
x^2+13x+42
show all the steps
To factorize the quadratic expression x^2 + 13x + 42, we will find two numbers that multiply to 42 (the constant term) and add up to 13 (the coefficient of the linear term, x).
Those two numbers are 6 and 7. You can see that 6*7 = 42 and 6+7 = 13.
So we can write the expression as:
x^2 + 6x + 7x + 42
Next, we group the terms and factor by grouping:
x(x + 6) + 7(x + 6)
Notice that both groups contain the term (x + 6), so we can factor that out to get:
(x + 6)(x + 7)
So the expression x^2 + 13x + 42 in factored form is (x + 6)(x + 7).
To find the expression in factored form, we need to factorize the given quadratic equation, which is x^2 + 13x + 42. Here are the steps to factorize it:
Step 1: Multiply the coefficient of the x^2 term (which is 1) and the constant term (which is 42). The result is 1 * 42 = 42.
Step 2: Find two numbers that multiply to give 42 (from step 1) and add up to the coefficient of the x term (which is 13). In this case, the two numbers are 6 and 7, because 6 * 7 = 42 and 6 + 7 = 13.
Step 3: Rewrite the original equation using the two numbers found in step 2. The equation becomes:
x^2 + 6x + 7x + 42.
Step 4: Group the terms in pairs:
(x^2 + 6x) + (7x + 42).
Step 5: Find the greatest common factor (GCF) for each pair of terms. In this case, the GCF of the first pair is x, and the GCF of the second pair is 7.
Step 6: Factor out the GCF from each pair of terms:
x(x + 6) + 7(x + 6).
Step 7: Notice that we now have a common binomial term, (x + 6), in both pairs. Factor this out:
(x + 6)(x + 7).
Therefore, the original expression in factored form is (x + 6)(x + 7).
To find the expression in factored form, we need to factorize the quadratic equation.
Step 1: Set the equation equal to zero:
x^2 + 13x + 42 = 0
Step 2: Find two numbers whose product is equal to the product of the coefficient of x^2 (1) and the constant term (42), and whose sum is equal to the coefficient of x (13).
In this case, the numbers are 6 and 7. (6 * 7 = 42 and 6 + 7 = 13)
Step 3: Rewrite the quadratic equation using these two numbers:
x^2 + 6x + 7x + 42 = 0
Step 4: Group the terms in pairs:
(x^2 + 6x) + (7x + 42) = 0
Step 5: Factor out the Greatest Common Factor (GCF) from each pair:
x(x + 6) + 7(x + 6) = 0
Step 6: Notice that both terms now have a common factor of (x + 6). Factor out (x + 6) from each term:
(x + 6)(x + 7) = 0
The factored form of the quadratic equation x^2 + 13x + 42 is (x + 6)(x + 7).