[1.] 2x² - 24x + 70
[2.] (m²+5m-14) / (m-2)
Answers
Step-by-step explanation:
(1) x2" was replaced by "x^2".
Step by step solution :
STEP 1: Equation at the end of step 1
(2x2 + 24x) + 70 = 0
STEP 2: Pulling out like terms
STEP 3 : Pull out like factors :
3.1 2x2 + 24x + 70 = 2 • (x2 + 12x + 35)
Trying to factor by splitting the middle term
3.2 Factoring x2 + 12x + 35
The first term is, x2 its coefficient is 1 .
The middle term is, +12x its coefficient is 12 .
The last term, "the constant", is +35
Step-1 : Multiply the coefficient of the first term by the constant 1 • 35 = 35
Step-2 : Find two factors of 35 whose sum equals the coefficient of the middle term, which is 12 .
-35 + -1 = -36
-7 + -5 = -12
-5 + -7 = -12
-1 + -35 = -36
1 + 35 = 36
5 + 7 = 12 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 5 and 7
x2 + 5x + 7x + 35
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+5)
Add up the last 2 terms, pulling out common factors :
7 • (x+5)
Step-5 : Add up the four terms of step 4 :
(x+7) • (x+5)
Which is the desired factorization
Equation at the end of step 3:
2 • (x + 7) • (x + 5) = 0
STEP 4 :
Theory - Roots of a product
4.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Equations which are never true:
4.2 Solve : 2 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Solving a Single Variable Equation:
4.3 Solve : x+7 = 0
Subtract 7 from both sides of the equation :
x = -7
Solving a Single Variable Equation:
4.4 Solve : x+5 = 0
Subtract 5 from both sides of the equation :
x = -5
(m + 2) • (m - 7) = 0
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