negative solution of the difference of the square of a number and 10 is equal to 3 times that number

Question

The difference of the square of a number and 10 is equal to 3 times that number. Find the negative solution.



Answer 1

x^2 -10 = 3x

difference means subtract 10 from x^2,

although it could mean 10-x^2

10-x^2 = 3x = 3 times the number

x^2 +3x -10 = 0

(x+5)(x-2) = 0

set each factor = 0 and solve for x

x+5=0

x=-5

x-2=0

x=2

odds are though that the difference between a number squared and ten means

x^2 -10 = 3x

x^2 -3x -10 = 0

(x-5)(x+2) = 0

x=5 or -2

But you could argue it either way. The english wording is a little ambiguous

the difference between 5 squared and 10 is 15 = 3 x 5

the difference between -2 squared and 10 is 4-10 = -6 = 3x -2

Versus

the differences with 10 minus the squared number

10-5 squared = 10 - 25 = -15 = 3 times -5

10- (2)^2 = 10-4 = 6 = 3 times +2

Answer is either -5 or -2 depending on how you translate English to math

If you get it wrong on a test or homework, I recommend arguing about it

Rocket scientists have miscalculated rocket projectories, due to using wrong units of

measurement. Answer keys and textbook solutions have occasional similar errors

In any event, the algebraic version of the English is a parabola. Graph it

and see where it intersects the x-axis. Those are the solutions

x^2 -3x + 10 is a parabola

y=f(x) = (x-3/2) ^2 + 31/4 with vertex at (3/2, 31/4)

It's zeroes are found by making y=0

0=x^2 -3x +10

x = 5,-2

but if you graph 10-x^2=3x

which is x^2+3x -10

you get x=--5, +2

as points where the parabola intersects the x-axi

Answer 2

First we translate the problem from English into algebra, where"x" is "a number":

The statement "the difference of the square of a number and 10 is equal to 3 times that number" is equivalent to the equation

x^2-10=3x .

Now, we subtract 3x from both sides of the equation in order to simplify it:

x^2-3x-10=0.

Next, we factor the quadratic on the left hand side:

(x-5)(x+2)=0.

Our two solutions are clearly 5 and -2. The negative solution is -2.