Questions on Word Problems: Mixtures answered by real tutors!

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How much pure water must be mixed with 3 pints of 70% developer to produce a
mixture that is 13% developer?
.
He started with:
3 pints of 70% developer
That means the amount of developer in the solution is:
.70 * 3 = 2.1 pints
.
Let x = amount of water added
then
2.1/(x+3) = .13(x+3)
2.1 = .13(x+3)^2
16.154 = (x+3)^2
16.154 = x^2 + 6x + 9
0 = x^2 + 6x - 7.154
.
Since you can't factor, you must apply the quadratic equation.
Doing so, produces two solutions one positive and one negative. We can throw out the negative solution cause it doesn't make sense. Therefore, the positive answer must be our solution:
x = 1.02 pints
.
Below is the solution of the quadratic equation:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation ax^2+bx+c=0

(in our case 1x^2+6x+-7.154 = 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=64.616 is greater than zero. That means that there are two solutions: $x[12] = (-6+-sqrt( 64.616 ))/2\a$ .

$x[1] = (-(6)+sqrt( 64.616 ))/2\1 = 1.01920390127199$
$x[2] = (-(6)-sqrt( 64.616 ))/2\1 = -7.01920390127199$

Quadratic expression 1x^2+6x+-7.154

can be factored:
1x+6x+-7.154 = 1(x-1.01920390127199)*(x--7.01920390127199)

Again, the answer is: 1.01920390127199, -7.01920390127199. Here's your graph:
graph( 500, 500, -10, 10, -20, 20, 1*x^2+6*x+-7.154 )