# How do you solve x^2-10x+9=0?

Dec 13, 2016

$x = 9$ and $x = 1$

#### Explanation:

To factor you must play with factors which multiply to $9$
$\left(1 \times 9 , \text{ "3xx3, " } 9 \times 1\right)$

$\left(x - 9\right) \left(x - 1\right) = 0$

Now, solve each factor for $0$:

$x - 9 = 0$

$x - 9 + 9 = 0 + 9$

$x - 0 = 9$

$x = 9$

and

$x - 1 = 0$

$x - 1 + 1 = 0 + 1$

$x - 0 = 1$

$x = 1$

Dec 13, 2016

$x = 1 \text{ }$ or $\text{ } x = 9$

#### Explanation:

Given:

${x}^{2} - 10 x + 9 = 0$

$\textcolor{w h i t e}{}$
Sum of coefficients shortcut

Notice that the sum of the coefficients is $0$.

That is:

$1 - 10 + 9 = 0$

Hence $x = 1$ is a solution and $\left(x - 1\right)$ a factor:

${x}^{2} - 10 x + 9 = \left(x - 1\right) \left(x - 9\right)$

There are several ways to spot that the other factor must be $\left(x - 9\right)$. For example, the coefficient of $x$ must be $1$ so that when multiplied by the $x$ in $\left(x - 1\right)$ results in ${x}^{2}$ and the constant term must be $- 9$ so that when multiplied by $- 1$ gives $+ 9$.

So the other solution is $x = 9$

$\textcolor{w h i t e}{}$
Completing the square

Another method, which is a little over the top for this particular problem, involves completing the square, then using the difference of squares identity:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

with $a = \left(x - 5\right)$ and $b = 4$ as follows:

$0 = {x}^{2} - 10 x + 9$

$\textcolor{w h i t e}{0} = {x}^{2} - 10 x + 25 - 16$

$\textcolor{w h i t e}{0} = {\left(x - 5\right)}^{2} - {4}^{2}$

$\textcolor{w h i t e}{0} = \left(\left(x - 5\right) - 4\right) \left(\left(x - 5\right) + 4\right)$

$\textcolor{w h i t e}{0} = \left(x - 9\right) \left(x - 1\right)$

Hence solutions $x = 9$ and $x = 1$

$\textcolor{w h i t e}{}$

For completeness, I should also mention the quadratic formula.

The equation:

${x}^{2} - 10 x + 9 = 0$

is in the form

$a {x}^{2} + b x + c = 0$

with $a = 1$, $b = - 10$ and $c = 9$

This has solutions given by the quadratic formula:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$\textcolor{w h i t e}{x} = \frac{10 \pm \sqrt{{\left(- 10\right)}^{2} - 4 \left(1\right) \left(9\right)}}{2 \cdot 1}$

$\textcolor{w h i t e}{x} = \frac{10 \pm \sqrt{100 - 36}}{2}$

$\textcolor{w h i t e}{x} = \frac{10 \pm \sqrt{64}}{2}$

$\textcolor{w h i t e}{x} = \frac{10 \pm 8}{2}$

$\textcolor{w h i t e}{x} = 5 \pm 4$

That is:

$x = 5 + 4 = 9 \text{ }$ or $\text{ } x = 5 - 4 = 1$