Find the real numbers.

Question

Find all real numbers  that satisfy the equation .

Answer 1

(x - 4)(x - 8) = 12

x² - 12x + 32 = 12

x² - 12x + 20 = 0

(x - 2)(x - 10) = 0

{2, 10}

Answer 2

Mark M. has the correct answer with the traditional method.

But sometimes it's useful to do some trial and error with a wider perspective.

If you multiply the two parenthetical factors, you get a quadratic expression, with highest degree 2.

That means you may have 2 answers. Start with some simple integers, such as 0, 1 and 2.

Let x=0, that gives 32 which does not equal 12.

try x=1, that gives 24 which is also too big

try x=2, that gives 12, so you have one answer, 2

try x=3, that gives 5 which is too small

x=4 gives zero,

x=5 gives a negative number, same with 6 and 7,

x=8 gives zero

x=9 gives 5

x=10 gives 12, so 10 is the other real answer

when you went from 3 to 4, the expression (x-4)(x-8) went from 5 to 0

from 2 to 3 to 4, it went from 12 to 5 to 0

from 8 to 9 to 10, it went from 0 to 5 to 12, just the reverse from 2 to 3 to 4

You could graph y=(x-4)(x-8) = x²-12x+32 It's a parabola with points (2,12) and (10,12)

x=2 and 10 give the necessary 12 value.

Or you could have taken x²-12x+32=12 and combined the constant terms to get

x²-12x+20=0 At that point you could solve it 3 ways

factor the left side and set each factor equal to zero, or

complete the square x-12x+36 +20-36 = add and subtract 1/2 the x coefficient squared (12/2)²=6²=36

that reduces to (x-6)²=16 take square roots of both sides to get x-6=+ or - 4 or x=2 or 10

or use the quadratic formula

or graph it and see where it intersects the x axis. It's a parabola that intersects the x-axis at x=2 and x=10

x²-12x+20 is a parabola with axis of symmetry where the first derivative equals zero 2x-12=0 or x=6

Once you had once answer x=2, the other solution is an equal distance to the right of the line of symmetry

6-2=4 6+4=10

Simplest way to solve it though was just factor x²-12x+20 into x-2 and x-10 then set each = 0

to get x-2=0 and x-10=0 solve each for x, to get x=2 and x=10.

The parabola crosses the x axis twice, for the two zeroes or two solutions, 2 and 10

Degree 2 equations have either 2, 1 or 0 real solutions, depending on whether the parabola crosses twice, touches only once or never touches the x-axis. If it touches just once, they call it repetitive solutions where both are the same value of x. If it never touches the x-axis, there are still two solutions, but they are imaginary, not real, involving i or the square root of negative one.

Sometimes it's useful to solve an equation more than one way. If you get the same answer both ways, you have more confidence you did it right.