factor 70 x^2+77 x+21 complex
To factor the quadratic expression 70x^2 + 77x + 21 over the complex numbers, we can follow these steps:
Step 1: Check if the expression can be factored by looking for common factors among the coefficients. In this case, there are no common factors other than 1.
Step 2: Since the coefficient of the x^2 term is not 1, we can't directly use the factoring methods for quadratic expressions of the form ax^2 + bx + c. Instead, we need to consider a different method.
Step 3: Look for two complex numbers, a and b, such that their sum is equal to the coefficient of the x term (77) and their product is equal to the product of the coefficient of the x^2 term (70) and the constant term (21).
In this case, we need to find complex numbers a and b such that a + b = 77 and a * b = 70 * 21 = 1470.
Step 4: To find the complex numbers a and b, we can solve the system of equations formed by the two conditions above.
Let's solve the equation a + b = 77 for one variable and substitute it into the other equation.
By isolating variable b in terms of a, we have b = 77 - a.
Substituting b = 77 - a into the equation a * b = 1470, we get: a * (77 - a) = 1470.
Expanding this equation, we have 77a - a^2 = 1470.
Rearranging the terms to form a quadratic equation, we get a^2 - 77a + 1470 = 0.
Step 5: Solve the quadratic equation a^2 - 77a + 1470 = 0 to find the complex numbers a and b.
We can factor this quadratic equation by finding two numbers whose sum is -77 and whose product is 1470. After factoring, we have (a - 42)(a - 35) = 0.
Setting each factor equal to zero, we get two possible values for a: a = 42 and a = 35.
Now, substitute these values of a back into the equation b = 77 - a to find the corresponding values of b:
When a = 42, b = 77 - 42 = 35.
When a = 35, b = 77 - 35 = 42.
So, the complex numbers a and b are a = 42 and b = 35, or a = 35 and b = 42.
Step 6: Now that we have found the complex numbers a and b, we can write the factored form of the quadratic expression.
The factored form is (x - a)(x - b).
Substituting the values of a and b, we get:
(x - 42)(x - 35).
Therefore, the factored form of the quadratic expression 70x^2 + 77x + 21 over the complex numbers is (x - 42)(x - 35).