Solve Quadratic equations 0=4x^3+25x^2-56x Tiger Algebra Solver (2024)

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

0-(4*x^3+25*x^2-56*x)=0

4.1 Pull out like factors:

-4x³ - 25x² + 56x=-x•(4x² + 25x - 56)

4.2Factoring 4x² + 25x - 56

The first term is, 4x² its coefficient is 4.
The middle term is, +25x its coefficient is 25.
The last term, "the constant", is -56

Step-1 : Multiply the coefficient of the first term by the constant 4•-56=-224

Step-2 : Find two factors of -224 whose sum equals the coefficient of the middle term, which is 25.

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -7 and 32
4x² - 7x+32x - 56

Step-4 : Add up the first 2 terms, pulling out like factors:
x•(4x-7)
Add up the last 2 terms, pulling out common factors:
8•(4x-7)
Step-5:Add up the four terms of step4:
(x+8)•(4x-7)
Which is the desired factorization

5.2Solve:-x = 0Multiply both sides of the equation by (-1) : x = 0

Root plot for : y = 4x²+25x-56
Axis of Symmetry (dashed) {x}={-3.12}
Vertex at {x,y} = {-3.12,-95.06}
x-Intercepts (Roots) :
Root 1 at {x,y} = {-8.00, 0.00}
Root 2 at {x,y} = { 1.75, 0.00}

6.2Solving4x²+25x-56 = 0 by Completing The Square.Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :
x²+(25/4)x-14 = 0

Add 14 to both side of the equation :
x²+(25/4)x = 14

Now the clever bit: Take the coefficient of x, which is 25/4, divide by two, giving 25/8, and finally square it giving 625/64

Add 625/64 to both sides of the equation :
On the right hand side we have:
14+625/64or, (14/1)+(625/64)
The common denominator of the two fractions is 64Adding (896/64)+(625/64) gives 1521/64
So adding to both sides we finally get:
x²+(25/4)x+(625/64) = 1521/64

Adding 625/64 has completed the left hand side into a perfect square :
x²+(25/4)x+(625/64)=
(x+(25/8))•(x+(25/8))=
(x+(25/8))²
Things which are equal to the same thing are also equal to one another. Since
x²+(25/4)x+(625/64) = 1521/64 and
x²+(25/4)x+(625/64) = (x+(25/8))²
then, according to the law of transitivity,
(x+(25/8))² = 1521/64

We'll refer to this Equation as Eq. #6.2.1

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

Note that the square root of
(x+(25/8))² is
(x+(25/8))^2/2=
(x+(25/8))¹=
x+(25/8)

Now, applying the Square Root Principle to Eq.#6.2.1 we get:
x+(25/8)= √ 1521/64

Subtract 25/8 from both sides to obtain:
x = -25/8 + √ 1521/64

Since a square root has two values, one positive and the other negative
x² + (25/4)x - 14 = 0
has two solutions:
x = -25/8 + √ 1521/64
or
x = -25/8 - √ 1521/64

Note that √ 1521/64 can be written as
√1521 / √64which is 39 / 8

6.3Solving4x²+25x-56 = 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= 25
C=-56 Accordingly,B²-4AC=
625 - (-896) =
1521Applying the quadratic formula :

-25 ± √ 1521
x=———————
8Can √ 1521 be simplified ?

Yes!The prime factorization of 1521is
3•3•13•13
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).

√ 1521 =√3•3•13•13 =3•13•√ 1 =
±39 •√ 1 =
±39

So now we are looking at:
x=(-25±39)/8

Two real solutions:

x =(-25+√1521)/8=(-25+39)/8= 1.750

or:

x =(-25-√1521)/8=(-25-39)/8= -8.000