Question

solve the equation for all real solutions in simplest form.
$d^2 - 12d - 14 = -6d$
answer attempt 1 out of 2
⊕ additional solution ⊖ no solution
$d = \square$

Explanation:

Step1: Rearrange to standard quadratic form

Bring all terms to left side:
$d^2 - 12d - 14 + 6d = 0$
Simplify:
$d^2 - 6d - 14 = 0$

Step2: Identify quadratic coefficients

For $ax^2+bx+c=0$, here:
$a=1$, $b=-6$, $c=-14$

Step3: Apply quadratic formula

Quadratic formula: $d=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute values:
$d=\frac{-(-6)\pm\sqrt{(-6)^2-4(1)(-14)}}{2(1)}$

Step4: Compute discriminant and simplify

Calculate discriminant:
$\sqrt{36 + 56}=\sqrt{92}=2\sqrt{23}$
Simplify numerator and denominator:
$d=\frac{6\pm2\sqrt{23}}{2}=3\pm\sqrt{23}$

Answer:

$d=3+\sqrt{23}$ and $d=3-\sqrt{23}$