One second degree equation is equation which can be written in the form ax2 + bx + c = 0. The letters The, B and ç represent real numbers constants called coefficients, and the coefficient a can never be equal to zero. When one of the other two coefficients, or both, is equal to zero, the equationofseconddegree formed is called incomplete.
So, the equationsincomplete can take one of the following three forms:
ax2 = 0
ax2 + bx = 0
ax2 + c = 0
each of these equations can be solved by techniques other than the Bhaskara's formula or by the method of to completesquares, which are unique in each of the three ways.
Bhaskara's formula
This is, without a doubt, the best known formula for solving equationsofseconddegree and can be used in any equation. As long as it has real solutions, the rootsreal of the equation will be obtained by this method, regardless of whether the equation is complete or incomplete. In fact, this formula can even be used to find solutions to equations that do not have real roots, in the set of complex numbers.
THE formulainBhaskara it is usually presented in two steps. So the first is the discriminating:
Δ = b2 – 4ac
And the second is:
x = – b ± √?
2nd
When the coefficientsB and C are equal to zero, we will have:
x = – b ± √(b2 – 4ac)
2nd
x = – 0 ± √(02 – 4th?·0)
2nd
x = 0
2nd
x = 0
So every time the coefficients B and C are equal to zero, we have discriminating equal to zero, so the equation will have only one real root. In this specific case, this result will be zero, as we found in the previous calculation.
When only the coefficient C = 0, we will have:
x = – b ± √(b2 – 4ac)
2nd
x = – b ± √(b2 – 4th?·0)
2nd
x = – b ± √(b2)
2nd
= – b ± b
2nd
This will result in x = 0 or x = b/a.
Do not stop now... There's more after the advertising ;)
When only the coefficient B = 0, we will have an equation with two real and distinct roots.
Alternative techniques for each type of equation
The techniques presented below are actually just an alternative that evades the use of Bhaskara's formula when the equations are incomplete. All of these calculations are based on the simple solution of equations and properties of mathematical operations.
When B and C are equal to zero
Just split the whole equation for the value of coefficient to and do the square root in both members of the equation. Note that the result will always be zero, as we will always have 0/a on the second member.
ax2 = 0
ax2 = 0
the a
x2 = 0
The
√x2 = √(0/a)
x = ± 0 = 0
When B = 0
If B is equal to zero, the procedure is the same as above, however, we must “pass” the term c/a to the second member before doing the square root on both members. Note that – c/a can be a positive number, as long as a or c is a negative number.
ax2 + c = 0
ax2 + ç = 0
a a a
ax2 = – ç
the a
x2 = - w/a
√x2 = ± √(– w/a)
Example:
2x2 – 50 = 0
2x2 = 50
x2 = 25
√x2 = √25
x = ± 5
When C = 0
If C = 0, we can put x in evidence:
ax2 + bx = 0
x (ax + b) = 0
As this is a product, one of the factors has to be zero for the equation is equal to zero. Therefore, x = 0 or:
ax + b = 0
ax = - b
x = - B
The
Example:
3x2 + 36 = 0
x (3x + 36) = 0
x = 0 or
3x + 36 = 0
3x = – 36
x = – 36
3
x = – 12
Hence, 0 and – 12 are the roots.
By Luiz Paulo Moreira
Graduated in Mathematics
Would you like to reference this text in a school or academic work? Look:
SILVA, Luiz Paulo Moreira. "What are incomplete second degree equations?"; Brazil School. Available in: https://brasilescola.uol.com.br/o-que-e/matematica/o-que-sao-equacoes-incompletas-segundo-grau.htm. Accessed on June 27, 2021.
Learn the definition of polynomial equation, define a polynomial function, the numerical value of a polynomial, the root or zero of the polynomial, Degree of a polynomial.
