Question

1. $3x^2 - 5x = 2$ →
2. $4x - 4 = -15x^2$ →
3. $3x^2 + 3 = 10x$ →
4. $2x^2 - 15 = -7x$ →
5. $6x^2 - 17x = -12$ →
6. $x^2 - 36 = 0$ →
7. $t^2 - 9 = 0$ →
8. $x^4 - 81 = 0$ →

Explanation:

Step1: Rewrite to standard quadratic form

For problem 59: $3x^2 -5x -2 =0$
For problem 60: $15x^2 +4x -4=0$
For problem 61: $3x^2 -10x +3=0$
For problem 62: $2x^2 +7x -15=0$
For problem 63: $6x^2 -17x +12=0$
For problem 64: $x^2 -36=0$
For problem 65: $t^2 -9=0$
For problem 66: $x^4 -81=(x^2-9)(x^2+9)=0$

Step2: Solve using quadratic formula

Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

1. 59: $x=\frac{5\pm\sqrt{25+24}}{6}=\frac{5\pm7}{6}$
2. 60: $x=\frac{-4\pm\sqrt{16+240}}{30}=\frac{-4\pm16}{30}$
3. 61: $x=\frac{10\pm\sqrt{100-36}}{6}=\frac{10\pm8}{6}$
4. 62: $x=\frac{-7\pm\sqrt{49+120}}{4}=\frac{-7\pm13}{4}$
5. 63: $x=\frac{17\pm\sqrt{289-288}}{12}=\frac{17\pm1}{12}$
6. 64: $x=\pm6$
7. 65: $t=\pm3$
8. 66: $x^2=9, x^2=-9$, so $x=\pm3, \pm3i$

Answer:

1. $\boldsymbol{x=2, \ x=-\frac{1}{3}}$
2. $\boldsymbol{x=\frac{2}{5}, \ x=-\frac{4}{3}}$
3. $\boldsymbol{x=3, \ x=\frac{1}{3}}$
4. $\boldsymbol{x=\frac{3}{2}, \ x=-5}$
5. $\boldsymbol{x=\frac{4}{3}, \ x=\frac{3}{2}}$
6. $\boldsymbol{x=6, \ x=-6}$
7. $\boldsymbol{t=3, \ t=-3}$
8. $\boldsymbol{x=3, \ x=-3, \ x=3i, \ x=-3i}$