# How do you solve 7x^2 - 4x - 1 = 0 using the quadratic formula?

Apr 6, 2016

$y = \frac{2 + \sqrt{11}}{7}$ or $y = \frac{2 - \sqrt{11}}{7}$

#### Explanation:

Using quadratic formula the solution of $a {x}^{2} + b x + c = 0$ is given by

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

As in $7 {x}^{2} - 4 x - 1 = 0$, $a = 7$, $b = - 4$ and $c = - 1$,

solution of $7 {x}^{2} - 4 x - 1 = 0$ is given by

$y = \frac{- \left(- 4\right) \pm \sqrt{{\left(- 4\right)}^{2} - 4 \cdot 7 \cdot \left(- 1\right)}}{2 \cdot 7}$

or $y = \frac{4 \pm \sqrt{16 + 28}}{14} = \frac{4 \pm \sqrt{44}}{14}$

Note that $\sqrt{44} = 2 \sqrt{11}$, hence

$y = \frac{4 \pm 2 \sqrt{11}}{14} = \frac{2 \pm \sqrt{11}}{7}$ i.e.

$y = \frac{2 + \sqrt{11}}{7}$ or $y = \frac{2 - \sqrt{11}}{7}$

Apr 6, 2016

$x = \frac{4 + \sqrt{11}}{7} , \frac{4 - \sqrt{11}}{7}$

#### Explanation:

color(blue)(7x^2-4x-1=0

This is a Quadratic equation (in form $a {x}^{2} + b x + c = 0$)

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

Remember that $a , b \mathmr{and} c$ are the coefficients

So,

color(purple)(a=7,b=-4,c=-1

$\rightarrow x = \frac{- \left(- 4\right) \pm \sqrt{- {4}^{2} - 4 \left(7\right) \left(- 1\right)}}{2 \left(7\right)}$

$\rightarrow x = \frac{4 \pm \sqrt{16 - \left(- 28\right)}}{14}$

$\rightarrow x = \frac{4 \pm \sqrt{16 + 28}}{14}$

$\rightarrow x = \frac{4 \pm \sqrt{44}}{14}$

$\rightarrow x = \frac{4 \pm \sqrt{4 \cdot 11}}{14}$

$\rightarrow x = \frac{4 \pm 2 \sqrt{11}}{14}$

$\rightarrow x = \frac{{\cancel{4}}^{2} \pm {\cancel{2}}^{1} \sqrt{11}}{{\cancel{14}}^{7}}$

$\rightarrow x = \frac{4 \pm \sqrt{11}}{7}$

Now we have two solutions

color(indigo)(x=(4+sqrt11)/7

color(violet)(x=(4-sqrt11)/(7)

color(blue)(ul bar |x=(4+sqrt11)/7,(4-sqrt11)/7|