How do you solve the quadratic equation 5x^2 + 3x = 1 ?

Aug 5, 2016

${x}_{1 , 2} = \frac{- 5 \pm \sqrt{13}}{10}$

Explanation:

The first thing to do here is rearrange your equation to quadratic form by getting all the terms on one side of the equation.

To do that, subtract $1$ from both sides of the equation

$5 {x}^{2} + 3 x - 1 = \textcolor{red}{\cancel{\textcolor{b l a c k}{1}}} - \textcolor{red}{\cancel{\textcolor{b l a c k}{1}}}$

$5 {x}^{2} + 3 x - 1 = 0$

Now, the quadratic formula allows you to calculate the two solutions of a general form quadratic equation

$\textcolor{b l u e}{a {x}^{2} + b x + c = 0}$

by using the equation

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {x}_{1 , 2} = \frac{- b \pm \sqrt{{b}^{2} - 4 \cdot a \cdot c}}{2 \cdot a} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\left\{\begin{matrix}a = \textcolor{w h i t e}{-} 5 \\ b = \textcolor{w h i t e}{-} 3 \\ c = - 1\end{matrix}\right.$

Plug these values into the quadratic formula to get

${x}_{1 , 2} = \frac{- 5 \pm \sqrt{{3}^{2} - 4 \cdot 1 \cdot \left(- 1\right)}}{2 \cdot 5}$

${x}_{1 , 2} = \frac{- 5 \pm \sqrt{13}}{10} \implies \left\{\begin{matrix}{x}_{1} = \frac{- 5 - \sqrt{13}}{10} \\ {x}_{2} = \frac{- 5 + \sqrt{13}}{10}\end{matrix}\right.$

You can thus say that your original equation has two solutions

$x = \frac{- 5 - \sqrt{13}}{10} \text{ }$ and $\text{ } x = \frac{- 5 + \sqrt{13}}{10}$

graph{5x^2 + 3x - 1 [-3, 3, -3, 3]}

Aug 5, 2016

$\frac{- 3 \pm \sqrt{29}}{10}$

Explanation:

Bring the quadratic equation to standard form:
$y = 5 {x}^{2} + 3 x - 1 = 0$
Use the new quadratic formula in graphic form:
$D = {d}^{2} = {b}^{2} - 4 a c = 9 + 20 = 29$ --> $d = \pm \sqrt{29}$
There are 2 real roots:
$x = - \frac{b}{2 a} \pm \frac{d}{2 a} = - \frac{3}{10} \pm \frac{\sqrt{29}}{10} = \frac{- 3 \pm \sqrt{29}}{10}$

NOTE. Using the improved quadratic formula gets simpler expressions and easier numeric computation. In addition, it shows students the graphic representation and interpretation of the axis of symmetry and the 2 x-intercepts.