Question

Supply the numbers to write x^2+10x+16=9 in its factored form. Enter the lesser number first.

Answer 1

To factor the equation \( x^2 + 10x + 16 = 9 \), we first rewrite it in standard form:

\[
x^2 + 10x + 16 - 9 = 0
\]

This simplifies to:

\[
x^2 + 10x + 7 = 0
\]

Next, we will factor \( x^2 + 10x + 7 \). To do this, we need two numbers that multiply to \( 7 \) (the constant term) and add to \( 10 \) (the coefficient of \( x \)).

The pairs of factors of \( 7 \) are \( (1, 7) \) and \( (-1, -7) \). However, none of these pairs add up to \( 10 \).

Since \( x^2 + 10x + 7 \) cannot be factored using integers, we can use the quadratic formula to find the roots of the equation:

\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\]

Here, \( a = 1, b = 10, c = 7 \):

\[
x = \frac{{-10 \pm \sqrt{{10^2 - 4 \cdot 1 \cdot 7}}}}{2 \cdot 1}
\]
\[
x = \frac{{-10 \pm \sqrt{{100 - 28}}}}{2}
\]
\[
x = \frac{{-10 \pm \sqrt{72}}}{2}
\]
\[
x = \frac{{-10 \pm 6\sqrt{2}}}{2}
\]
\[
x = -5 \pm 3\sqrt{2}
\]

Thus, the two roots are:

\[
x_1 = -5 + 3\sqrt{2}, \quad x_2 = -5 - 3\sqrt{2}
\]

Now, the factored form can be written using these roots:

\[
x^2 + 10x + 7 = (x - (-5 + 3\sqrt{2}))(x - (-5 - 3\sqrt{2}))
\]

So, the factored form is:

\[
(x + 5 - 3\sqrt{2})(x + 5 + 3\sqrt{2})
\]

If you need just the lesser of the two roots as requested, it is:

\[
-5 - 3\sqrt{2}
\]

Typically, for factoring, you would not express it in terms of irrational numbers, but in context, if used, you would put negative values first. The numbers used are:

\[
-5 - 3\sqrt{2}, -5 + 3\sqrt{2}
\]

So the lesser number first when writing is

\[
-5 - 3\sqrt{2}, -5 + 3\sqrt{2}
\]

And if you want the concise representation, it is:

\(-5 - 3\sqrt{2}, -5 + 3\sqrt{2}\).