-190b=-105-25bsquared what is that using binomials
-190b = -105 - 25b^2.
25b^2 - 190b + 105 = 0
Divide both sides by 5:
5b^2 - 38b + 21 = 0.
Solve using Quadratic Formula and get:
b = 7.
b = 0.6.
Thanks alot man
To solve the equation using binomials, we can rearrange the terms to form a quadratic equation.
Starting with -190b = -105 - 25b^2, we can rewrite it as:
25b^2 + 190b - 105 = 0
Now, let's express the quadratic equation in the standard form, which is ax^2 + bx + c = 0, where a, b, and c are constants.
In our case:
a = 25
b = 190
c = -105
To factor the quadratic equation, we need to find two binomials in the form of (m + n)(p + q) that will multiply together to give us the quadratic equation.
We look for two numbers, let's say m and n, such that their sum (m + n) equals the coefficient of the linear term (b), which is 190.
Next, we find two numbers, let's say p and q, such that their product (p * q) equals the product of the coefficient of the quadratic term (a) and the constant term (c), which is 25 * -105 = -2625.
Let's find these pairs of numbers:
We can break down 190 into factors: 5*38 and 2*5*19.
For -2625, some factor pairs are: -1*2625, -3*875, -5*525, -7*375, -15*175, -25*105, -35*75, -105*25.
After examining these pairs, we see that -35 * 75 equals -2625, and -35 + 75 equals 40, which matches the linear term (b = 190).
Now, we'll rewrite the quadratic equation by splitting the linear term using these numbers:
25b^2 - 35b + 75b - 105 = 0
Next, we group the terms:
(25b^2 - 35b) + (75b - 105) = 0
Now, let's factor out common terms from each grouping:
5b(5b - 7) + 15(5b - 7) = 0
We can observe that we have a common binomial expression, (5b - 7), in both groupings.
So, we can factor it out:
(5b - 7)(5b + 15) = 0
Now, we can use the zero product property, which states that if a product of factors is equal to zero, then at least one of the factors must be equal to zero.
Setting each factor to zero and solving for b:
5b - 7 = 0 or 5b + 15 = 0
When solving the first equation:
5b = 7
b = 7/5
And when solving the second equation:
5b = -15
b = -15/5
b = -3
Therefore, the solutions to the given equation using binomials are b = 7/5 and b = -3.