Why are there usually two solutions to quadratic equations? Please provide an example.?

Answer 1

Looking at the quadratic formula, there is a part called the 'discriminant'. This is the b^2 - 4ac under the sqrt sign. Square roots always have positive and negative values, so there are two roots except in the case where the discriminant is exactly equal to zero. And if the disciminant is negative you still have two roots, but they are imaginary.

Visually, think of the parabola that forms the equation. If you draw a horizontal line through it, it will almost always cross at two points. The only exception is if it crosses at the vertex. The roots are the places where the line x = 0 (x-axis) cross through the graph.
If it crosses through the parabola --> two real roots (discriminant is positive)
If it crosses through the vertex --> one real root (discriminant is zero)
If it doesn't cross the parabola --> two complex roots (discriminant is negative)

Answer 2

The best way to understand this is to remember the graph of a quadratic is a parabola, and the zeros come where the parabola crosses the X axis (where y = 0). If the parabola does not touch the X axis there are no 0's, if the vertex is on it there is 1, otherwise it hits it in 2 places, and there are 2 zeros.

Another way to think of it is by factoring. You might not know this yet, but a quadratic is equal to 0 only if it can be written as the product of 2 linear factors: (ax + b)(cx + d). I am sure you do know that a product of 2 factors is 0 only if one or both of the factor is 0. Clearly the liear factors each has a solution. If the quadratic does not factor there are no solutions. There is just one solution if the quadratic is a perfect square, that is (ax + b)(ax + b).

Example?
x^2 - 2x - 8 = (x - 4)(x + 2) has 2 zeros, at 4 and -2.
x^2 - 2x + 1 = (x - 1)(x - 1) has 1 zero, at x = 1
x^2 - 2x + 8 does not factor and has no zeros

Or, easier to see:
x^2 - 1 = y has 2 zeros, at +- 1
x^2 = y has 1 zero, at 0
x^2 + 1 = y has no zeros.

Answer 3

It's quite difficult to provide an example of this but i'll give it a try. If you have a quadratic for example- x2 (squared) +3x+5 and sketch it on a graph by substituting in different values for x for example, -3,-2,-1,0,1,2,3 then the shape on the graph will represent a smile like this >>>>>>U. This U shape will cut the x axis twice as it goes down then back up again (the u shape) so there are 2 solutions for x. The two solutions are where the graph cuts the x axis. =) hope that was a help & was at least a little bit understandable!

Answer 4

Use substitution. From the first equation you can derive y = 7-x, then substitue that for y in the second equation, giving x^2 + (7-x)^2 = 85 Expand x^2 + 49 -14x + x^2 = 85 Combine terms 2x^2 - 14x - 36 = 0 then solve the quadratic for x.

Answer 5

The best way to understand this is to note that a quadratic equation, Ax^2+Bx+C = 0 can be written in the form,

x=[ -B +- sqrt(B^2-4AC) ] / 2A

which gives the solutions,

x= -B+ sqrt(.......

and

x= -B - sqrt(........

so there are 2 solutions. in the case where the B^2-4AC part is less than zero, we have the square root of a negative number which is not defined as a real number. (it can however be defined as a complex number but you might not need to know this yet!)

Answer 6

Just as Kuji said.
For example x^2 - 1 can be expanded as (x-1)(x+1)
and if we look at it
y=(x-1)(x+1)
Two answers also means the points of intersection of the curve with the x-axis!

x^2 - 1 = 0
(x-1)(x+1) = 0
x= -1 or x = 1
when y = 0, x =1 or x =-1, which means the curve cuts the axis at those points

Answer 7

The fundamental theorem of algebra (proved by Gauss) proves that:
every complex polynomial p(z) in one variable and of degree n ≥ 1 has some complex root. In other words, the field of complex numbers is algebraically closed and therefore, as for any other algebraically closed field, the equation p(z) = 0 has n roots (not necessarily distinct).

for quadratic equation n = 2, so the number of roots(solutions) is 2

See the link for the proofs of the theorem

Answer 8

What Kuji said. Notice that -2 squared = 4 and 2 squared = 4. Another way of putting it is the square root of 4 is +/-2.

Answer 9

There are usually 2 solutions because any number is the square of both a pair of numbers and their negatives.

Answer 10

in a qwadratic equation x is raised to the power of 2 so it has two solution. if the power of x be 3 or more there would surely be 3 ans. & more. no. of ans depends on the power of variable.
got it dear!!!!!!!!!!! ex x *x+4x+2