### explain General formula for quadratic equation with examples?

explain General formula for quadratic equation with examples?

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This is the general quadratic equation formula. We define it as follows: If ax2 + bx + c = 0 is a quadratic equation, then the value of x is given by the following formula: Just plug in the values of a, b and c, and do the calculations. The quantity in the square root is called the discriminant or D.
Example 1: Solve the equation: x2 + 3x – 4 = 0
Solution: This method is also known as splitting the middle term method. Here, a = 1, b = 3, c = -4. Let us multiply a and c = 1 * (-4) = -4. Next, the middle term is split into two terms. We do it such that the product of the new coefficients equals the product of a and c.

We have to get 3 here. Consider (+4) and (-1) as the factors, whose multiplication is -4 and sum is 3. Hence, we write x2 + 3x – 4 = 0 as x2 + 4x – x – 4 = 0. Thus, we can factorise the terms as: (x+4)(x-1) = 0. For any two quantities a and b, if a×b = 0, we must have either a = 0, b = 0 or a = b = 0.

Thus we have either (x+4) = 0 or (x-1) = 0 or both are = 0. This gives x+4 = 0 or x-1 = 0. Solving these equations for x gives: x=-4 or x=1. This method is convenient but is not applicable to every equation. In those cases
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ax²+ bx +c=0
this is the general quadratic formula
equations in that form is solved in four ways
1 factorization method
2. formula method
3 completing of square method
4. graphical metho
x²-7x+ 10
using method (1) factorization
sum of roots= -7x
priduct of root = +10x²
therefore, the root can be found by trial and error method,
now find two nōs that when added gives -7x and when multipied gives +10x²
i.e -5 and -2
x²-5x-2x+ 10
by parring
(x²-5x)(-2x+10)
factorise
x( x -5) -2( x- 5)
therefore,
(x-2) ( x-5)
casei
x-2=0
x= 2
caseii
x-5=0
x= 5
x= 2,5
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In elementary algebra, the quadratic formula is a formula which provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others.

Given a general quadratic equation of the form

{\displaystyle ax^{2}+bx+c=0} ax^2+bx+c=0
with {\displaystyle x} x representing an unknown, {\displaystyle a} a, {\displaystyle b} b and {\displaystyle c} c representing constants with {\displaystyle a\neq 0} a\neq 0, the quadratic formula is:

{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\ \ }
{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\ \ }
where the plus-minus symbol "±" indicates that the quadratic equation has two solutions. Written separately, they become:

Here, each of these two solutions is also called a root of the quadratic equation (in fact, one can verify that each solution satisfies the quadratic equation by inserting them into the equation itself). Geometrically, these roots represent the {\displaystyle x} x values at which any parabola, explicitly given as {\displaystyle y=ax^{2}+bx+c} {\displaystyle y=ax^{2}+bx+c}, crosses the {\displaystyle x} x-axis.

As well as being a formula that yields the zeros of any parabola, the quadratic formula can also be used to identify the axis of symmetry of the parabola, and the number of real zeros the quadratic equation contains.

Equivalent formulations Edit
The quadratic formula may also be written as:

{\displaystyle x={\frac {-b}{2a}}\pm {\sqrt {\frac {b^{2}-4ac}{4a^{2}}}}\ \ ,} {\displaystyle x={\frac {-b}{2a}}\pm {\sqrt {\frac {b^{2}-4ac}{4a^{2}}}}\ \ ,}
which may be simplified to:

{\displaystyle x=-{\frac {b}{2a}}\pm {\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}}}\ \ .} {\displaystyle x=-{\frac {b}{2a}}\pm {\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}}}\ \ .}
This version of the formula is convenient when complex roots are involved, in which case the expression outside the square root will be the real part—and the square root expression the imaginary part. The expression inside the square root is a discriminant
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A+bx+c=0
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