Question

This question is based on your work on MU123 up to and including Unit 10.

(a) This part of the question concerns the quadratic equation 4x² +6x+2=0.
(i) Find the discriminant of the quadratic expression 4x² +6x+2.
(ii) What does this discriminant tell you about the number of solutions of the equation? Explain your answer briefly. (iii) What does this discriminant tell you about the graph of y=4x² +6x+2?

Answer 1

The discriminant of the quadratic equation 4x² +6x+2=0 is positive (4), indicating 2 real and distinct solutions. The graph of y=4x² +6x+2 opens upwards and intersects the x-axis at two distinct points.

Step-by-step explanation:

The quadratic equation given is 4x² +6x+2=0. Here are the steps to answer the questions:

1. (i) The discriminant of the quadratic expression 4x² +6x+2 is calculated using the formula, discriminant = b² - 4ac. In this case, a = 4, b = 6, and c = 2. Substituting these values, we get discriminant = (6)² - 4(4)(2) = 36 - 32 = 4.
2. (ii) The discriminant tells us about the number of solutions of the equation. If the discriminant is positive, there are 2 real and distinct solutions. If the discriminant is zero, there is 1 real and repeated solution. If the discriminant is negative, there are no real solutions. In this case, the discriminant is positive (4), so there are 2 real and distinct solutions.
3. (iii) The discriminant also tells us about the graph of the quadratic equation. If the discriminant is positive, the graph of the equation opens upwards and intersects the x-axis at 2 distinct points. If the discriminant is zero, the graph touches the x-axis at one point. If the discriminant is negative, the graph does not intersect the x-axis. In this case, since the discriminant is positive (4), the graph of y=4x² +6x+2 opens upwards and intersects the x-axis at two distinct points.