(4y)square -(23)square
Answers
Answer:
Step by step solution :
STEP
1
:
Equation at the end of step 1
(22y2 - 23y) + 15 = 0
STEP
2
:
Trying to factor by splitting the middle term
2.1 Factoring 4y2-23y+15
The first term is, 4y2 its coefficient is 4 .
The middle term is, -23y its coefficient is -23 .
The last term, "the constant", is +15
Step-1 : Multiply the coefficient of the first term by the constant 4 • 15 = 60
Step-2 : Find two factors of 60 whose sum equals the coefficient of the middle term, which is -23 .
-60 + -1 = -61
-30 + -2 = -32
-20 + -3 = -23 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -20 and -3
4y2 - 20y - 3y - 15
Step-4 : Add up the first 2 terms, pulling out like factors :
4y • (y-5)
Add up the last 2 terms, pulling out common factors :
3 • (y-5)
Step-5 : Add up the four terms of step 4 :
(4y-3) • (y-5)
Which is the desired factorization
Equation at the end of step
2
:
(y - 5) • (4y - 3) = 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 : y-5 = 0
Add 5 to both sides of the equation :
y = 5
Solving a Single Variable Equation:
3.3 Solve : 4y-3 = 0
Add 3 to both sides of the equation :
4y = 3
Divide both sides of the equation by 4:
y = 3/4 = 0.750
Supplement : Solving Quadratic Equation Directly
Solving 4y2-23y+15 = 0 directly
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula
Parabola, Finding the Vertex:
4.1 Find the Vertex of t = 4y2-23y+15