The square of a number exceeds the number by 72. Find the number

Answer 1

We're told that the square of a number exceeds that same number by 72.

Math word problems always have certain keywords that help you set up the equation. In this case the keyword is 'exceeds', which signifies addition. So we set the problem up as such, using X to represent the number.

X² = X +72

To solve for X, we will have to move everything to one side of the equals sign, leaving a 0 behind. Subtract (X+72) from both sides. This will be our starting equation.

X² - X - 72 = 0

This is a quadratic equation, expressed in the form of

aX² + bX + c = 0, or

(pX + q)(rX + s) = 0, or

pX + q=0, and rX + s = 0.

Quadratic equations can have at most two solutions.

Look at the coefficients of our starting equation. In a quadratic formula, q and s have to add up to b, and multiply to equal c.

The first term is X². The coefficient a is just 1. So the p and the r of the underlined equation have to equal 1 when multiplied together. So they are each either 1 or -1. We'll start with 1 for now and come back to -1 to check.

The second term is -X. The coefficient b is -1. So q and s have to add up to -1.

The third term is -72. So q and s have to equal -72 when multiplied together.

It is generally understood that when a=1, trial and error is the fastest way to solve, rather than using the "complete the square" method. So let's find the q and s numbers that are equal to -72 when multiplied and -1 when added together.

q s multiplied added

1 -72 -72 -71

2 -36 -72 -34

3 -24 -72 -21

4 -18 -72 -14

we're approaching -1 fast, so keep going

5 (no integers multiplied by 5 will give you -72)

6 -12 -72 -6

7 (no integers multiplied by 7 will give you -72)

8 -9 -72 -1

q = 8 and s = -9. So,

pX+8 =0 and rX-9=0. We said that since p and r had to equal 1 when multiplied together, we'd assume they were both equal to one. In that case:

X+8=0 and X-9=0, so

X=-8 or X=9,

Try substituting -8 in the original equation.

X² = X + 72

(-8)² = (-8) + 72

64 = -8 +72

64 = 64. Yes, the square of -8, which is 64, is 72 greater than -8 itself.

Try 9.

(9)² = (9) +72

81 = 81. Yes, the square of 9, which is 81, is 72 greater than 9 itself.

Both -8 and 9 are correct.

Circling back around to -1 as promised (even though we know quadratics can only have two solutions at the most, and we just solved both of them)

if p = -1 and r = -1, the equations simplify to

-X + 8 = 0 and -X - 9 = 0

and X will equal 8 or -9.

(8)² = 8 + 72

64 = 80 false

(-9)² = -9 + 72

81 = 63 false

Answer 2

We will do x^2=x+72. This equation represents the fact that the square of a respective number x is greater than than same number by 72.

Solve for x and you will get that number.

Let's do it.

Let's subtract x+72 from the right side and get, x^2-x-72=0.

Now, let's factor this. We get (x-9)(x+8)=0. Thus, our x's are

x=-8 and 9.

The answer to this question cannot be a negative number. So, 9 has to be the answer.