# Question e94e0

Apr 27, 2017

$r = 7 \mathmr{and} r = 2$

#### Explanation:

color(blue)(r^2=-14+9r

Bring everything to the left hand side

$\rightarrow {r}^{2} - 9 r + 14 = 0$

Now, this is a quadratic equation. Solve it by using the Quadratic formula

color(brown)(x=(-b+-sqrt(b^2-4ac))/(2a)

Where $x$ is the variable $r$ and $a , b \mathmr{and} c$ are the coefficients of the terms $\left(a = 1 , b = - 9 \mathmr{and} c = 14\right)$

$\rightarrow r = \frac{- \left(- 9\right) \pm \sqrt{{\left(- 9\right)}^{2} - 4 \left(1\right) \left(14\right)}}{2 \left(1\right)}$

$\rightarrow r = \frac{9 \pm \sqrt{81 - 56}}{2}$

$\rightarrow r = \frac{9 \pm \sqrt{25}}{2}$

$\rightarrow r = \frac{9 \pm 5}{2}$

Now, there are two solutions for $r$

$\textcolor{p u r p \le}{r = \frac{9 + 5}{2} = \frac{14}{2} = 7}$

$\textcolor{v i o \le t}{r = \frac{9 - 5}{2} = \frac{4}{2} = 2}$

color(green)( :.r=7 color(green)(or color(green)(2#

Hope this helps..! :)

Apr 27, 2017

$r = 2 \mathmr{and} r = 7$

#### Explanation:

Re-arrange it into general form first: $a {x}^{2} + b x + c = 0$

It does not matter at all what the variable is!

${r}^{2} - 9 r + 14 = 0$

This means: $a = 1 \text{ "b = -9" } c = 14$

The quadratic formula is: $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Use the values for $a , b , c$ above and substitute.

$r = \frac{- \left(- 9\right) \pm \sqrt{{\left(- 9\right)}^{2} - 4 \left(1\right) \left(14\right)}}{2 \left(1\right)} \text{ } \leftarrow$ simplify

$r = \frac{9 \pm \sqrt{\left(81 - 56\right)}}{2}$

There are two possible answers for $r$

$r = \frac{9 + \sqrt{25}}{2} = 7$

$r = \frac{9 - \sqrt{25}}{2} = 2$

We could also have solved the original equation by finding the factors.

Apr 27, 2017

$r = 2 \text{ or } r = 7$

#### Explanation:

$\text{rearrange and equate to zero}$

$\Rightarrow {r}^{2} - 9 r + 14 = 0$

$\text{compare to the standard form } a {x}^{2} + b x + c = 0$

$\text{here } a = 1 , b = - 9 , c = 14$

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

$\textcolor{w h i t e}{\Rightarrow r} = \frac{- \left(- 9\right) \pm \sqrt{{\left(- 9\right)}^{2} - \left(4 \times 1 \times 14\right)}}{2}$

$\textcolor{w h i t e}{\Rightarrow r} = \frac{9 \pm \sqrt{81 - 56}}{2} = \frac{9 \pm \sqrt{25}}{2}$

$\Rightarrow r = \frac{9 + 5}{2} = 7 \text{ or } r = \frac{9 - 5}{2} = 2$