# How do you solve y = -2x^2 + 5x - 1?

May 11, 2018

The roots are:

$\left(\frac{5 + \sqrt{17}}{4} , 0\right)$ and $\left(\frac{5 - \sqrt{17}}{4} , 0\right)$

The approximate roots are:

$\left(0.2192 , 0\right)$ and $\left(2.281 , 0\right)$

#### Explanation:

Solve:

$y = - 2 {x}^{2} + 5 x - 1$ is a quadratic equation in standard form:

$y = a {x}^{2} + b x + c$,

where:

$a = - 2$, $b = 5$, $c = - 1$

Substitute $0$ for $y$.

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

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

Plug in the known value.

$x = \frac{- 5 \pm \sqrt{{5}^{2} - 4 \cdot - 2 \cdot - 1}}{2 \cdot - 2}$

Simplify.

$x = \frac{- 5 \pm \sqrt{25 - 8}}{- 4}$

$x = \frac{- 5 \pm \sqrt{17}}{- 4}$

Simplify.

$x = \frac{5 \pm \sqrt{17}}{4}$

$x = \frac{5 + \sqrt{17}}{4} ,$ $\frac{5 - \sqrt{17}}{4}$

The roots are:

$\left(\frac{5 + \sqrt{17}}{4} , 0\right)$ and $\left(\frac{5 - \sqrt{17}}{4} , 0\right)$

The approximate roots are:

$\left(0.2192 , 0\right)$ and $\left(2.281 , 0\right)$

graph{y=-2x^2+5x-1 [-10, 10, -5, 5]}

May 11, 2018

$x = - 0.22 \text{ and } x = 2.28$ for $y = 0$

#### Explanation:

When we say "solution" instead of the entire curve, we usually mean the place(s) were the function is zero. That is, the "roots" of the equation are at:

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

Now you can solve this be number of ways - factoring, quadratic formula, and graphing.

This one may be solved most quickly with the quadratic formula.

x =( −b ± sqrt(b^2−4ac))/(2a)
in this case, $a = - 2 , b = 5 , c = - 1$

x =( −5 ± sqrt(5^2−4(-2)(-1)))/(2(-2))

x = (−5 ± sqrt(17))/(−4)

x = 5/4 ± sqrt(17)/4 ; x = 5/4 ± 1.03

$x = - 0.22 \text{ and } x = 2.28$