solve for t.
3t-18=4(-3-3√4t)
Answers
Step-by-step explanation:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
3*t-18-(4*(-3-3*t/4*t))=0
Step by step solution :
STEP
1
:
t
Simplify —
4
Equation at the end of step
1
:
t
(3t - 18) - (4 • (-3 - ((3 • —) • t))) = 0
4
STEP
2
:
Rewriting the whole as an Equivalent Fraction :
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using 4 as the denominator :
-3 -3 • 4
-3 = —— = ——————
1 4
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
2.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
-3 • 4 - (3t2) -3t2 - 12
—————————————— = —————————
4 4
Equation at the end of step
2
:
(-3t2 - 12)
(3t - 18) - (4 • ———————————) = 0
4
STEP
3
:
STEP
4
:
Pulling out like terms
4.1 Pull out like factors :
-3t2 - 12 = -3 • (t2 + 4)
Polynomial Roots Calculator :
4.2 Find roots (zeroes) of : F(t) = t2 + 4
Polynomial Roots Calculator is a set of methods aimed at finding values of t for which F(t)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers t which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is 4.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 5.00
-2 1 -2.00 8.00
-4 1 -4.00 20.00
1 1 1.00 5.00
2 1 2.00 8.00
4 1 4.00 20.00
Polynomial Roots Calculator found no rational roots
Equation at the end of step
4
:
(3t - 18) - -3 • (t2 + 4) = 0
STEP
5
:
STEP
6
:
Pulling out like terms
6.1 Pull out like factors :
3t2 + 3t - 6 = 3 • (t2 + t - 2)
Trying to factor by splitting the middle term
6.2 Factoring t2 + t - 2
The first term is, t2 its coefficient is 1 .
The middle term is, +t its coefficient is 1 .
The last term, "the constant", is -2
Step-1 : Multiply the coefficient of the first term by the constant 1 • -2 = -2
Step-2 : Find two factors of -2 whose sum equals the coefficient of the middle term, which is 1 .
-2 + 1 = -1
-1 + 2 = 1 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 2
t2 - 1t + 2t - 2
Step-4 : Add up the first 2 terms, pulling out like factors :
t • (t-1)
Add up the last 2 terms, pulling out common factors :
2 • (t-1)
Step-5 : Add up the four terms of step 4 :
(t+2) • (t-1)
Which is the desired factorization
Equation at the end of step
6
:
3 • (t + 2) • (t - 1) = 0
STEP
7
:
Theory - Roots of a product
7.1 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:
7.2 Solve : 3 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Solving a Single Variable Equation:
7.3 Solve : t+2 = 0
Subtract 2 from both sides of the equation :
t = -2
Solving a Single Variable Equation:
7.4 Solve : t-1 = 0
Add 1 to both sides of the equation :
t = 1
Supplement : Solving Quadratic Equation Directly
Solving t2+t-2 = 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
Answer: