- 5x² + 4x - 45 midiltarm
Answers
Answer:
2.1 Factoring 5x2+34x+45
The first term is, 5x2 its coefficient is 5 .
The middle term is, +34x its coefficient is 34 .
The last term, "the constant", is +45
Step-1 : Multiply the coefficient of the first term by the constant 5 • 45 = 225
Step-2 : Find two factors of 225 whose sum equals the coefficient of the middle term, which is 34 .
-225 + -1 = -226
-75 + -3 = -78
-45 + -5 = -50
-25 + -9 = -34
-15 + -15 = -30
-9 + -25 = -34
-5 + -45 = -50
-3 + -75 = -78
-1 + -225 = -226
1 + 225 = 226
3 + 75 = 78
5 + 45 = 50
9 + 25 = 34 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 9 and 25
5x2 + 9x + 25x + 45
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (5x+9)
Add up the last 2 terms, pulling out common factors :
5 • (5x+9)
Step-5 : Add up the four terms of step 4 :
(x+5) • (5x+9)
Which is the desired factorization
Step-by-step explanation:
Equation at the end of step
2
:
(5x + 9) • (x + 5) = 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 : 5x+9 = 0
Subtract 9 from both sides of the equation :
5x = -9
Divide both sides of the equation by 5:
x = -9/5 = -1.800
Solving a Single Variable Equation:
3.3 Solve : x+5 = 0
Subtract 5 from both sides of the equation :
x = -5
Supplement : Solving Quadratic Equation Directly
Solving 5x2+34x+45 = 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 y = 5x2+34x+45
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 5 , is positive (greater than zero).
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