# How do you solve x^2-8x+5=0?

Mar 15, 2018

We use a process called completing the square, which works for all quadratic equations. Here it gives $x = 4 \setminus \pm \setminus \sqrt{11}$.

#### Explanation:

Rather than memorizing a formula, you should understand how "completing the square" works.

Render the equation as:

${x}^{2} - 8 x = - 5$

Then add a constant, which we will call ${b}^{2}$:

$\textcolor{b l u e}{{x}^{2} - 8 x + {b}^{2} = {b}^{2} - 5}$

Now compare the left side with the identity

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

We then have

${x}^{2} - 8 x + {b}^{2} = {\left(a + b\right)}^{2}$

if

${a}^{2} = {x}^{2}$ so set $a = x$

$2 a b = - 8 x$ so $b = - 4$

So,go back to the equation in blue and put in $b = - 4$, then:

${x}^{2} - 8 x + 16 = {\left(a + b\right)}^{2} = {\left(x - 4\right)}^{2} = 11$

So $x - 4 = \setminus \pm \setminus \sqrt{11}$ meaning

$x = 4 \setminus \pm \setminus \sqrt{11}$.

Mar 15, 2018

Root 1 color(blue)( = 4+sqrt(11)

Root 2 color(blue)( = 4-sqrt(11)

#### Explanation:

Given:

color(red)(x^2-8x+5=0

The standard form of a quadratic is color(blue)(ax^2+bx+c=0

So, we have color(blue)(a=1, b=-8 and c=5.

Solutions are given by color(red)([-b+-sqrt(b^2-4ac)]/(2a)

$\Rightarrow \frac{- \left(- 8\right) \pm \sqrt{{\left(- 8\right)}^{2} - 4 \left(1\right) \left(5\right)}}{2 \cdot 1}$

Using scientific calculator, we get,

rArr [8+-sqrt(44)]/(2 [ Intermediate Result 1 ]

Observe that $\sqrt{44}$ can also be simplified as

$\sqrt{4 \cdot 11}$

$\Rightarrow \sqrt{{2}^{2} \cdot 11}$

$\Rightarrow \sqrt{{2}^{2}} \cdot \sqrt{11}$

$\Rightarrow 2 \sqrt{11}$

Now, we can write [ Intermediate Result 1 ] as

$\textcolor{b l u e}{\frac{8 \pm 2 \sqrt{11}}{2}}$

We can rewrite the above as

$\textcolor{b l u e}{\frac{8}{2} \pm \frac{2 \sqrt{11}}{2}}$

$\frac{4 \cdot 2}{2} \pm \frac{2 \sqrt{11}}{2}$

$\frac{4 \cdot \cancel{2}}{\cancel{2}} \pm \frac{\cancel{2} \sqrt{11}}{\cancel{2}}$

$\Rightarrow 4 \pm \sqrt{11}$

Hence, we have the following two solutions:

Root 1 color(blue)( = 4+sqrt(11)

Root 2 color(blue)( = 4-sqrt(11)

Using a scientific calculator, we can simplify the above results.

Root 1 color(green)(~~ 7.31662479

Root 2color(green)(~~ 0.68337521

You can verify the solutions visually by examining the image of the graph below:

Hope it helps.