Questions on Algebra: Quadratic Equation answered by real tutors!

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Find two real numbers that have a sum of 8 and a product of 2.
.
Let x = one of two real numbers
and y - second of two real numbers
.
Since we have two unknowns, we'll need two equations.
.
From:"that have a sum of 8" we get equation 1:
x + y = 8
.
From:"product of 2" we get equation 2:
xy = 2
.
Solving equation 1 for y:
x + y = 8
y = 8-x
.
Plug the above into equation 2 and solve for x:
xy = 2
x(8-x) = 2
8x-x^2 = 2
0 = x^2 - 8x + 2
Since, we can't factor, we must use the quadratic equation.
x = {7.742, 0.258}
y = (0.258, 7.742)
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Answer: the two numbers are 7.742 and 0.258
.
Details of quadratic solution follows:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation ax^2+bx+c=0

(in our case 1x^2+-8x+2 = 0

) has the following solutons:

$x[12] = (b+-sqrt( b^2-4ac ))/2\a$

For these solutions to exist, the discriminant b^2-4ac

should not be a negative number.

First, we need to compute the discriminant b^2-4ac

.

Discriminant d=56 is greater than zero. That means that there are two solutions: $x[12] = (--8+-sqrt( 56 ))/2\a$ .

$x[1] = (-(-8)+sqrt( 56 ))/2\1 = 7.74165738677394$
$x[2] = (-(-8)-sqrt( 56 ))/2\1 = 0.258342613226059$

Quadratic expression 1x^2+-8x+2

can be factored:
1x+-8x+2 = 1(x-7.74165738677394)*(x-0.258342613226059)

Again, the answer is: 7.74165738677394, 0.258342613226059. Here's your graph:
graph( 500, 500, -10, 10, -20, 20, 1*x^2+-8*x+2 )