Solve Quadratic equations x^2-x+756=0 Tiger Algebra Solver (2024)

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.

For tidiness, printing of 42 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Root plot for : y = x²-x+756
Axis of Symmetry (dashed) {x}={ 0.50}
Vertex at {x,y} = { 0.50,755.75}
Function has no real roots

2.2Solvingx²-x+756 = 0 by Completing The Square.Subtract 756 from 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 -3023/4
So adding to both sides we finally get:
x²-x+(1/4) = -3023/4

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) = -3023/4 and
x²-x+(1/4) = (x-(1/2))²
then, according to the law of transitivity,
(x-(1/2))² = -3023/4

We'll refer to this Equation as Eq. #2.2.1

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.#2.2.1 we get:
x-(1/2)= √ -3023/4

Add 1/2 to both sides to obtain:
x = 1/2 + √ -3023/4
In Math,iis called the imaginary unit. It satisfies i²=-1. Both i and -i are the square roots of -1

Since a square root has two values, one positive and the other negative
x² - x + 756 = 0
has two solutions:
x = 1/2 + √ 3023/4 • i
or
x = 1/2 - √ 3023/4 • i

Note that √ 3023/4 can be written as
√3023 / √4which is √3023 / 2

2.3Solvingx²-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 =
-3023Applying the quadratic formula :

1 ± √ -3023
x=——————
2In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written (a+b*i)

Both i and -i are the square roots of minus 1

Accordingly,√-3023=
√3023•(-1)=
√3023•√-1=
±√ 3023 •i

√ 3023 , rounded to 4 decimal digits, is 54.9818
So now we are looking at:
x=(1± 54.982 i )/2

Two imaginary solutions :