Quadratic Formula & Discriminant

Front

What is the Quadratic Formula?

Back

The Quadratic Formula is used to find the solutions of a quadratic equation in the form ax² + bx + c = 0. It is given by: x = (-b ± √(b² - 4ac)) / (2a).

Front

What does the discriminant of a quadratic equation indicate?

Back

The discriminant (D = b² - 4ac) indicates the nature of the roots of the quadratic equation. If D > 0, there are 2 distinct real solutions; if D = 0, there is 1 real solution; if D < 0, there are no real solutions.

Front

How do you calculate the discriminant for the equation 4x² + 4x + 1 = 0?

Back

For the equation 4x² + 4x + 1, a = 4, b = 4, c = 1. The discriminant is D = b² - 4ac = 4² - 4(4)(1) = 16 - 16 = 0.

Front

What is the significance of a discriminant equal to 0?

Back

A discriminant equal to 0 indicates that the quadratic equation has exactly one real solution, also known as a repeated or double root.

Front

How do you find the maximum height of a projectile?

Back

The maximum height of a projectile can be found using the vertex formula t = -b/(2a) from the quadratic equation h(t) = at² + bt + c.

Front

What is the vertex of a parabola?

Back

The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens upwards or downwards.

Front

Given h(t) = -16t² + 128t, when does the rocket reach maximum height?

Back

To find when the rocket reaches maximum height, use t = -b/(2a). Here, a = -16 and b = 128, so t = -128/(2*-16) = 4 seconds.

Front

What is the height of a projectile at time t = 0?

Back

At time t = 0, the height is given by h(0) = c, where c is the initial height. For h(t) = -16t² + 128t, h(0) = 0.

Front

How do you determine the number of solutions for the equation -8x² + 6x - 4 = 0?

Back

Calculate the discriminant: D = b² - 4ac = 6² - 4(-8)(-4) = 36 - 128 = -92. Since D < 0, there are no real solutions.

Front

What is the formula for the height of an object thrown upwards?

Back

The height of an object thrown upwards can be modeled by the equation h(t) = -16t² + vt + h₀, where v is the initial velocity and h₀ is the initial height.