Solve Quadratic equations x^2-x-756=0 Tiger Algebra Solver (2025)

1.1Factoring x²-x-756

The first term is, x² its coefficient is 1.
The middle term is, -x its coefficient is -1.
The last term, "the constant", is -756

Step-1 : Multiply the coefficient of the first term by the constant 1•-756=-756

Step-2 : Find two factors of -756 whose sum equals the coefficient of the middle term, which is -1.

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -28 and 27
x² - 28x+27x - 756

Step-4 : Add up the first 2 terms, pulling out like factors:
x•(x-28)
Add up the last 2 terms, pulling out common factors:
27•(x-28)
Step-5:Add up the four terms of step4:
(x+27)•(x-28)
Which is the desired factorization

Root plot for : y = x²-x-756
Axis of Symmetry (dashed) {x}={ 0.50}
Vertex at {x,y} = { 0.50,-756.25}
x-Intercepts (Roots) :
Root 1 at {x,y} = {-27.00, 0.00}
Root 2 at {x,y} = {28.00, 0.00}

3.2Solvingx²-x-756 = 0 by Completing The Square.Add 756 to both side of the equation :
x²-x = 756

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:
756+1/4or, (756/1)+(1/4)
The common denominator of the two fractions is 4Adding (3024/4)+(1/4) gives 3025/4
So adding to both sides we finally get:
x²-x+(1/4) = 3025/4

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) = 3025/4 and
x²-x+(1/4) = (x-(1/2))²
then, according to the law of transitivity,
(x-(1/2))² = 3025/4

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)= √ 3025/4

Add 1/2 to both sides to obtain:
x = 1/2 + √ 3025/4

Since a square root has two values, one positive and the other negative
x² - x - 756 = 0
has two solutions:
x = 1/2 + √ 3025/4
or
x = 1/2 - √ 3025/4

Note that √ 3025/4 can be written as
√3025 / √4which is 55 / 2

3.3Solvingx²-x-756 = 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= 1
B= -1
C=-756 Accordingly,B²-4AC=
1 - (-3024) =
3025Applying the quadratic formula :

1 ± √ 3025
x=——————
2Can √ 3025 be simplified ?

Yes!The prime factorization of 3025is
5•5•11•11
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).

√ 3025 =√5•5•11•11 =5•11•√ 1 =
±55 •√ 1 =
±55

So now we are looking at:
x=(1±55)/2

Two real solutions:

x =(1+√3025)/2=(1+55)/2= 28.000

or:

x =(1-√3025)/2=(1-55)/2= -27.000