Find the zeros of quadratic polynomial 7x2 – 6x – 16.
Answers
Answer:
7x2+6x-16=0
Two solutions were found :
x = -2
x = 8/7 = 1.143
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(7x2 + 6x) - 16 = 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 7x2+6x-16
The first term is, 7x2 its coefficient is 7 .
The middle term is, +6x its coefficient is 6 .
The last term, "the constant", is -16
Step-1 : Multiply the coefficient of the first term by the constant 7 • -16 = -112
Step-2 : Find two factors of -112 whose sum equals the coefficient of the middle term, which is 6 .
-112 + 1 = -111 -56 + 2 = -54 -28 + 4 = -24 -16 + 7 = -9 -14 + 8 = -6 -8 + 14 = 6 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -8 and 14
7x2 - 8x + 14x - 16
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (7x-8)
Add up the last 2 terms, pulling out common factors :
2 • (7x-8)
Step-5 : Add up the four terms of step 4 :
(x+2) • (7x-8)
Which is the desired factorization
Equation at the end of step 2 :
(7x - 8) • (x + 2) = 0
Step 3 :
Theory - Roots of a product :
3.1 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.
Solving a Single Variable Equation :
3.2 Solve : 7x-8 = 0
Add 8 to both sides of the equation :
7x = 8
Divide both sides of the equation by 7:
x = 8/7 = 1.143
Solving a Single Variable Equation :
3.3 Solve : x+2 = 0
Subtract 2 from both sides of the equation :
x = -2
Answer: