Solve the quadratic equation -5m^2 + 12m + 8 = 3 - 5m, giving your answers to 2 decimal places.

Step-by-step solution

1. 1

   Move everything to the side that keeps the squared term positive. Adding $5m^2$ and subtracting $12m+8$ from both sides sends the terms right; equivalently, collect on the left and then multiply by $-1$:

   $-5m^{2}+12m+8-3+5m=0\ \Longrightarrow\ -5m^{2}+17m+5=0\ \Longrightarrow\ 5m^{2}-17m-5=0.$

   A positive leading coefficient makes the signs in the quadratic formula much easier to keep straight.

2. 2

   Read off the coefficients. $a=5$, $b=-17$, $c=-5$. Note that $c$ is negative, so $-4ac$ will be positive and add to the discriminant.

3. 3

   Compute the discriminant.

   $b^{2}-4ac=(-17)^{2}-4(5)(-5)=289+100=389.$

   $389$ is prime, so the surd cannot be simplified and the roots must be given as decimals.

4. 4

   Apply the quadratic formula.

   $m=\frac{17\pm\sqrt{389}}{10}.$

5. 5

   Evaluate to 2 decimal places. With $\sqrt{389}=19.72308\ldots$,

   $m=\frac{17+19.72308}{10}=3.672308\ldots\approx 3.67,\qquad m=\frac{17-19.72308}{10}=-0.272308\ldots\approx -0.27.$

6. 6

   Check with Vieta's formulas. For $5m^2-17m-5=0$ the roots should sum to $\tfrac{17}{5}=3.4$ and multiply to $\tfrac{-5}{5}=-1$. Indeed $3.672308+(-0.272308)=3.4$ and $3.672308\times(-0.272308)=-1.0000$, so both roots are right.