Range of parameter k for real and same-sign roots

Hi Jerismen, let's solve this quadratic equation problem together. We are given the equation k minus three times x squared plus four x plus k equals zero, and we know it has real roots.

For part A, we need to find the range of values for k. Since the equation has real roots, the discriminant must be greater than or equal to zero.

Let's identify our coefficients. A is k minus three, B is four, and C is k.

Substituting these into our discriminant formula, we get four squared minus four times k minus three times k.

Let's expand the terms. Sixteen minus four k times the quantity k minus three is greater than or equal to zero.

Distributing the negative four, we have sixteen minus four k squared plus twelve k is greater than or equal to zero.

Next, let's divide the entire inequality by negative four. Remember, when dividing by a negative number, we must flip the inequality sign.

Now we factor the quadratic expression on the left. We need two numbers that multiply to negative four and add to negative three. Those are negative four and positive one.