Solve Quadratic equations 4x^2-4x-5=0 Tiger Algebra Solver (2024)

2.1Factoring 4x²-4x-5

The first term is, 4x² its coefficient is 4.
The middle term is, -4x its coefficient is -4.
The last term, "the constant", is -5

Step-1 : Multiply the coefficient of the first term by the constant 4•-5=-20

Step-2 : Find two factors of -20 whose sum equals the coefficient of the middle term, which is -4.

Root plot for : y = 4x²-4x-5
Axis of Symmetry (dashed) {x}={ 0.50}
Vertex at {x,y} = { 0.50,-6.00}
x-Intercepts (Roots) :
Root 1 at {x,y} = {-0.72, 0.00}
Root 2 at {x,y} = { 1.72, 0.00}

3.2Solving4x²-4x-5 = 0 by Completing The Square.Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :
x²-x-(5/4) = 0

Add 5/4 to both side of the equation :
x²-x = 5/4

Now the clever bit: Take the coefficient of x, which is 1, divide by two, giving 1/2, and finally square it giving 1/4

Add 1/4 to both sides of the equation :
On the right hand side we have:
5/4+1/4The common denominator of the two fractions is 4Adding (5/4)+(1/4) gives 6/4
So adding to both sides we finally get:
x²-x+(1/4) = 3/2

Adding 1/4 has completed the left hand side into a perfect square :
x²-x+(1/4)=
(x-(1/2))•(x-(1/2))=
(x-(1/2))²
Things which are equal to the same thing are also equal to one another. Since
x²-x+(1/4) = 3/2 and
x²-x+(1/4) = (x-(1/2))²
then, according to the law of transitivity,
(x-(1/2))² = 3/2

We'll refer to this Equation as Eq. #3.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
(x-(1/2))² is
(x-(1/2))^2/2=
(x-(1/2))¹=
x-(1/2)

Now, applying the Square Root Principle to Eq.#3.2.1 we get:
x-(1/2)= √ 3/2

Add 1/2 to both sides to obtain:
x = 1/2 + √ 3/2

Since a square root has two values, one positive and the other negative
x² - x - (5/4) = 0
has two solutions:
x = 1/2 + √ 3/2
or
x = 1/2 - √ 3/2

Note that √ 3/2 can be written as
√3 / √2

3.3Solving4x²-4x-5 = 0 by the Quadratic Formula.According to the Quadratic Formula,x, the solution forAx²+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= -4
C= -5 Accordingly,B²-4AC=
16 - (-80) =
96Applying the quadratic formula :

4 ± √ 96
x=—————
8Can √ 96 be simplified ?

Yes!The prime factorization of 96is
2•2•2•2•2•3
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).

√ 96 =√2•2•2•2•2•3 =2•2•√ 6 =
±4 •√ 6

√ 6 , rounded to 4 decimal digits, is 2.4495
So now we are looking at:
x=(4±4• 2.449 )/8

Two real solutions:

x =(4+√96)/8=(1+√ 6 )/2= 1.725

or:

x =(4-√96)/8=(1-√ 6 )/2= -0.725