Question

3÷3+1-1÷2=2÷3x-1 into quad eq
​

Answer 1

Answer:

mark me as brainliest and follow me

Answer 2

Answer:

Step-by-step explanation:

can be used to derive a general formula for solving quadratic equations, called the quadratic formula.[5] The mathematical proof will now be briefly summarized.[6] It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:

${\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.}\left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.$

Taking the square root of both sides, and isolating x, gives:

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}$

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax2 + 2bx + c = 0 or ax2 − 2bx + c = 0 ,[7] where b has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.

A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.

A lesser known quadratic formula, as used in Muller's method provides the same roots via the equation

${\displaystyle x={\frac {2c}{-b\pm {\sqrt {b^{2}-4ac}}}}.}{\displaystyle x={\frac {2c}{-b\pm {\sqrt {b^{2}-4ac}}}}.}$

This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is c/a.

One property of this form is that it yields one valid root when a = 0, while the other root contains division by zero, because when a = 0, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form 0/0 for the other root. On the other hand, when c = 0, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form 0/0.

______________________________________________

additional information

Reduced quadratic equation

It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by a, which is always possible since a is non-zero. This produces the reduced quadratic equation:[8]

${\displaystyle x^{2}+px+q=0,}x^{2}+px+q=0,$

where p = b/a and q = c/a. This monic equation has the same solutions as the original.

The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is:

${\displaystyle x={\frac {1}{2}}\left(-p\pm {\sqrt {p^{2}-4q}}\right),}{\displaystyle x={\frac {1}{2}}\left(-p\pm {\sqrt {p^{2}-4q}}\right),}$

or equivalently:

${\displaystyle x=-{\frac {p}{2}}\pm {\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}.}{\displaystyle x=-{\frac {p}{2}}\pm {\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}.}$

hope it helps