solve the equation 4x square + 12 x minus 7 is equal to zero by completing the square method
Answers
Answer:
(4x)2+12×(-7)=0
16xsquare+12×(-7)
16xsquare+(-84)=0
16xsquare-84=0
16xsquare=0+84
x=84÷16
x=21÷4
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(22x2 + 12x) - 7 = 0
Step 2 :
Trying to factor by splitting the middle term
Factoring 4x2+12x-7
The first term is, 4x2 its coefficient is 4 .
The middle term is, +12x its coefficient is 12 .
The last term, "the constant", is -7
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -2 and 14
4x2 - 2x + 14x - 7
Step-4 : Add up the first 2 terms, pulling out like factors :
2x • (2x-1)
Add up the last 2 terms, pulling out common factors :
7 • (2x-1)
Step-5 : Add up the four terms of step 4 :
(2x+7) • (2x-1)
Which is the desired factorization
Equation at the end of step 2 :
(2x - 1) • (2x + 7) = 0
Step 3 :
Theory - Roots of a product :
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 :
Solve : 2x-1 = 0
Add 1 to both sides of the equation :
2x = 1
Divide both sides of the equation by 2:
x = 1/2 = 0.500
Solving a Single Variable Equation :
Solve : 2x+7 = 0
Subtract 7 from both sides of the equation :
2x = -7
Divide both sides of the equation by 2:
x = -7/2 = -3.500
Supplement : Solving Quadratic Equation Directly
Solving 4x2+12x-7 = 0 directly
Solving 4x2+12x-7 = 0 by Completing The Square .
Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :
x2+3x-(7/4) = 0
Add 7/4 to both side of the equation :
x2+3x = 7/4
Add 9/4 to both sides of the equation :
On the right hand side we have :
7/4 + 9/4 The common denominator of the two fractions is 4 Adding (7/4)+(9/4) gives 16/4
So adding to both sides we finally get :
x2+3x+(9/4) = 4
Adding 9/4 has completed the left hand side into a perfect square :
x2+3x+(9/4) =
(x+(3/2)) • (x+(3/2)) =
(x+(3/2))2
Things which are equal to the same thing are also equal to one another. Since
x2+3x+(9/4) = 4 and
x2+3x+(9/4) = (x+(3/2))2
then, according to the law of transitivity,
(x+(3/2))2 = 4
We'll refer to this Equation as Eq. #4.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x+(3/2))2 is
(x+(3/2))2/2 =
(x+(3/2))1 =
x+(3/2)
x+(3/2) = √ 4
Subtract 3/2 from both sides to obtain:
x = -3/2 + √ 4
Since a square root has two values, one positive and the other negative
x2 + 3x - (7/4) = 0
has two solutions:
x = -3/2 + √ 4
or
x = -3/2 - √ 4
Solve Quadratic Equation using the Quadratic Formula
4.3 Solving 4x2+12x-7 = 0 by the Quadratic Formula .
According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
x = ————————
2A
In our case, A = 4
B = 12
C = -7
Accordingly, B2 - 4AC =
144 - (-112) =
256
Applying the quadratic formula :
-12 ± √ 256
x = ——————
8
Can √ 256 be simplified ?
Yes! The prime factorization of 256 is
2•2•2•2•2•2•2•2
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).
√ 256 = √ 2•2•2•2•2•2•2•2 =2•2•2•2•√ 1 =
± 16 • √ 1 =
± 16
So now we are looking at:
x = ( -12 ± 16) / 8
Two real solutions:
x =(-12+√256)/8=-3/2+2= 0.500
or:
x =(-12-√256)/8=-3/2-2= -3.500