For an easy 10 points:?

Question

Tell me how the quadratic formula is made?
I need EVERY step to give a person credit.
I want to see who knows how the quadratic formula was invented.

This ain't no homework or nuthin I know the answer.
I'll give you a hint, use the completing the square method.

Change ax2+by+c=0 to (-b+/-Square root of (b^2-4ac))/2a

Answer 1

x^2+bx+c = 0

x^2 +bx/a+c/a =0

x^2+bx/a= - c/a

x^2+bx/a+ b^2 / 4a^2 = b^2 / 4a^2 -c/a (this is completing the square step)

(x+b/2a)^2 = (b^2-4ac) / (4a^2)

therefore

x+b/2a = + or - sqrt(b^2-4ac) / 2a

x = ((-b) + or - sqrt(b^2-4ac)) / 2a

Answer 2

Theory:
A quadratic equation looks like this:

ax² + bx + c = 0 (where ‘a’ cannot be zero.)

Solving the equation means finding ‘x’ values that make the equation true. These ‘x’ values are called the roots of the quadratic.

Quadratic equations can have 0, 1 or two roots.

NOTE: In the complex number system, all quadratic equations have roots, but we will not discuss complex numbers in this article. Roots of quadratics always come in pairs, but when there are two roots that are the same we say that there is only one root.

The quadratic formula is derived from the general quadratic equation (below) by completing the square.

The general quadratic equation...

ax² + bx + c = 0

has roots...



This formula, known as the ‘quadratic formula’, is actually two formulas. The ‘±’ symbol should be read as ‘plus or minus’, which means that you have to work out the formula twice, once with a plus sign in that position, then again with a minus sign.

The first step is to identify the coefficients ‘a’, ‘b’ and ‘c’ in your quadratic equation, so that you can substitute them into the formula to calculate ‘x’.

For this equation:

x² - 4x - 5 = 0

There is no number written in front of the x² term, but in that case it is helpful to think of the x² term as 1x² , so then:

a = 1, b = -4, and c = -5

Substituting these values into the formula we get:



NOTE: If the expression under the square root sign is negative, then there are no real roots and you cannot go any further. You can investigate this before you start by calculating: b² - 4ac

Answer 3

the quadratic formula comes from solving for x in the standard form of a quadratic equation. you must use completing the square.
1) ax^2 + bx + c = 0

2) divide everything by a => x^2 + (b/a) x + (c/a) = 0

3) subtract c/a from each side => x^2 + (b/a) x = - (c/a)

4) divide the coeff of x by 2, square this and add it to both sides of the equation => x^2 + (b/a) x + (b^2 / 4a^2) = (b^2 / 4a^2) -(c/a) {that was the completing the square part}

5) use your rules for factoring special products =>
( x + (b/2a)) ^ 2 = ( b^2 - 4ac) / (4a^2)

6) take the square root of both sides =>
( x + (b/2a)) = sqrt(b^2 - 4ac) / 2a

7) subtract b/2a from both sides {this is the quadratic formula} =>
x = (- b + - sqrt(b^2 - 4ac) )/ 2a

this doesn't tell you how it was invented but it tells from where it was derived.

Answer 4

(m)

This is the quadratic formula, as it is taught to most of us in school:

x1,2=(-b/2a) ± (1/2a)(b2-4ac)1/2

gives the solution to a generic quadratic equation of the form:

ax2 + bx + c = 0

The development, or derivation, of a mathematical idea is usually as logical, deducible and rectilinear as possible. This brings about the common notion that its historical development is similarly as continuous, logical and rectilinear: one mathematician picking up an idea where another mathematician left it.

Using the quadratic formula as an example, it will be shown that the historical development of mathematics is not at all rectilinear. Instead, parallel developments, interconnections and confluences can be found, which - to complicate this stuff even further - are also interrelated with social, cultural, political and religious matters.

The so-called quadratic formula has been derived in the course of a few millennia to its current form, which is taught to most of us in school. This Entry will strictly concentrate on the historical development of the quadratic formula. Some mathematical background may be of use to fully understand the described development, however the maths used in this Entry will be kept at a necessary minimum.

Answer 5

The Bablonians were the first to used them, in recorded history between 1800 and 1600 BC. Also Chinese and Babylonians used a method for finding square for positive roots, but did not yet have a formula.

The Bakshali Manuscript of India showed an algebraic formula for solving quadratic in the positive.

The first mathematician to have found negative solutions with the general algebraic formula, was Brahmagupta around 600. Muhammad around 800 developed a set of formulae that worked for positives. Abraham bar Hiyya Ha-Nasi introduced the complete solution to Europe in 1100s. Bhaskara II of India solved quadratic equations with more than one unknown also in the 1100s.

Shridhara of India 9th century was one of the first mathematicians to give a general rule for solving the equation. His work is lost but Bhaskara II later quotes Shridhara's rule:

"Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root. "

Answer 6

I think it was developed from this thing called completing the square.

You set f(x) to = zero
Then you get the xs on the left of the = and the constants on the right hand side
Then you take 1/2 the coefficent of x, square it and add it to both sides of the equation.
Factor the left hand side
Solve by taking the square root of both sides.

Heres an example

f(x) = x^2+6x-17
x^2+6X-17=0
x^2+6x=17
x^2+6x+(3)^2=17+(3)^2
x^2+6x+9=26
(x+3)(x+3)=26
(x+3)^2=26
x+3=+-square root of 26
x=+-square root of 26-3

somehow it was developed into the quadratic formula

Answer 7

I know it was invented by someone in prison. It states that

x=the opposite of b plus or minus the square root of b squared minus 4ac all divided by 2a

when an equation is in the form of ax^2+bx+c=0

Answer 8

That is so NOT worth ten points. If you really want to know, go buy a book on algebraic proofs or something. The origin of the quadratic formula is probably pages long.

Answer 9

it's x= -b +/- the square root of b squared -4ac divided by 2a...i seem to remember in my crazy math class learning that this formula was invented in BC not sure when but that the babylonians first used it so it's got to be really old.

Answer 10

Quadratic formula is...

x = negative b plus/minus square root of b squared minus 4ac all over 2a

I don't know who invented it though.

Answer 11

Ax(cubed)+Bx(squared)+Cx+D=0