Question

1. solve the equation for all real solutions in simplest form

$k^2 - 2k - 11 = 3k$

1. solve the equation for all real solutions in simplest form.

$6k^2 + 3k - 4 = 2k^2$

1. solve the equation for all real solutions in simplest form.

$5v^2 - 3v + 1 = 4v$

1. solve the equation for all real solutions in simplest form.

$5b^2 + 3b - 4 = -1$

1. solve the equation for all real solutions in simplest form

$3t^2 + 12t + 14 = 4$

1. solve the equation for all real solutions in simplest form.

$4d^2 + 4d + 1 = -6d$

1. solve the equation for all real solutions in simplest form.

$3k^2 + 2k - 3 = 0$

1. solve the equation for all real solutions in simplest form.

$10b^2 - 12b + 3 = 5b^2$

1. solve the equation for all real solutions in simplest form.

$-4p^2 + 8p + 3 = -5p^2$
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Explanation:

Let's solve question 18: \( 3k^2 + 2k - 3 = 0 \) using the quadratic formula.

Step 1: Identify coefficients

For a quadratic equation \( ax^2 + bx + c = 0 \), here \( a = 3 \), \( b = 2 \), \( c = -3 \).

Step 2: Quadratic formula

The quadratic formula is \( k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Step 3: Calculate discriminant

Discriminant \( D = b^2 - 4ac = (2)^2 - 4(3)(-3) = 4 + 36 = 40 \).

Step 4: Substitute into formula

\( k = \frac{-2 \pm \sqrt{40}}{2(3)} = \frac{-2 \pm 2\sqrt{10}}{6} = \frac{-1 \pm \sqrt{10}}{3} \).

Answer:

\( k = \frac{-1 + \sqrt{10}}{3}, \frac{-1 - \sqrt{10}}{3} \)