Question

solve the equation for all real solutions in simplest form.
$11w^2 - 11w + 3 = 6w^2$
answer attempt 1 out of 2
additional solution no solution
$w = $

Explanation:

Step1: Rearrange to standard quadratic form

Subtract $6w^2$ from both sides:
$11w^2 - 6w^2 - 11w + 3 = 0$
$5w^2 - 11w + 3 = 0$

Step2: Identify coefficients for quadratic formula

For $aw^2 + bw + c = 0$, we have $a=5$, $b=-11$, $c=3$.

Step3: Apply quadratic formula

Quadratic formula: $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Substitute values:
$w = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(5)(3)}}{2(5)}$

Step4: Calculate discriminant and simplify

Compute discriminant:
$\sqrt{121 - 60} = \sqrt{61}$
Simplify numerator and denominator:
$w = \frac{11 \pm \sqrt{61}}{10}$

Answer:

$w = \frac{11 + \sqrt{61}}{10}$ and $w = \frac{11 - \sqrt{61}}{10}$