The square of a positive number is 56 more than the number itself. What is the number?

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The square of a positive number is 56 more than the number itself. What is the number?

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Answer 1 · 2017-01-09T21:53:54

The number is $8$ $8$

Explanation:

We need to take this one phrase at a time to develop our equation.

First, the square of a positive number can be written as:

$x^{2}$ $x^2$

In mathematics the word "is"means "=" so we can now write:

$x^{2} =$ $x^2 =$

and "56 more than the number itself" finishes the equation as:

$x^{2} = 56 + x$ $x^2 = 56 + x$

We can now transform this into a quadratic:

$x^{2} - 56 - x = 56 + x - 56 - x$ $x^2 - color(red)(56 - x) = 56 + x - color(red)(56 - x)$

$x^{2} - x - 56 = 0$ $x^2 - x - 56 = 0$

We can now factor the quadratic:

$(x - 8) (x + 7) = 0$ $(x - 8)(x + 7) = 0$

Now we can solve each term for $0$ $0$

$x + 7 = 0$ $x + 7 = 0$

$x + 7 - 7 = 0 - 7$ $x + 7 - 7 = 0 - 7$

$x + 0 = - 7$ $x + 0 = -7$

$x = - 7$ $x = -7$ - this cannot be the answer because the question asked for a positive integer.

$x - 8 = 0$ $x - 8 = 0$

$x - 8 + 8 = 0 + 8$ $x - 8 + 8 = 0 + 8$

$x - 0 = 8$ $x - 0 = 8$

$x = 8$ $x = 8$

The number is $8$ $8$

Answer 2 · 2017-01-09T22:06:59

$8$ $8$

Explanation:

Let the unknown value be $x$ $x$

This is a quadratic in disguise.

$x^{2} = x + 56 \Rightarrow x^{2} - x - 56 = 0$ $x^2=x+56" "=>" "x^2color(red)(-x)-56=0$

The $x$ $color(red)(x)$ has the coefficient of -1. This means that the whole number factors of 56 have a difference of -1.

$\sqrt{56} \approx 7.5$ $sqrt(56)~~7.5$

Try $(- 8) \times (+ 7) = - 56 and 7 - 8 = - 1$ $(-8)xx(+7) =-56 " and "7-8=-1$ so we have found the factors

$x^{2} - x - 56 = (x - 8) (x + 7) = 0$ $x^2-x-56=(x-8)(x+7)=0$

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The question stipulates that the number is positive so we select $x = + 8$ $x=+8$
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$Check$ $color(blue)("Check")$

$x^{2} = x + 56 \to 8^{2} \to 8 + 56$ $x^2=x+56" "->" "8^2->8+56$

$64 \to 64$ $" "64->64$

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The square of a positive number is 56 more than the number itself. What is the number?

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