Question

determining solutions to quadratic equations
the solution to ( x^2 - 10x = 24 ) is ( \boldsymbol{12} ) or ( \boldsymbol{-2} )
complete
the solution to ( 2x^2 - 11 = 87 ) is ( \boldsymbol{7} ) or ( \boldsymbol{-7} )
complete
the solution to ( 3x^2 - 12x + 24 = 0 ) is
done
( 7 ) or ( -7 )
( 2 + 2i ) or ( 2 - 2i )
( 12 ) or ( -2 )

Explanation:

Step1: Simplify the quadratic equation

Divide all terms by 3:
$$x^2 - 4x + 8 = 0$$

Step2: Identify coefficients for quadratic formula

For $ax^2+bx+c=0$, $a=1$, $b=-4$, $c=8$

Step3: Apply quadratic formula

Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute values:
$$x=\frac{4\pm\sqrt{(-4)^2-4(1)(8)}}{2(1)}$$

Step4: Calculate discriminant and simplify

Compute discriminant: $\sqrt{16-32}=\sqrt{-16}=4i$
Simplify expression:
$$x=\frac{4\pm4i}{2}=2\pm2i$$

Answer:

$2 + 2i$ or $2 - 2i$