Quadratic formula and discriminant

Front

What is the quadratic formula?

Back

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the solutions of a quadratic equation ax² + bx + c = 0.

Front

What does the discriminant tell us?

Back

The discriminant (D = b² - 4ac) indicates the nature of the roots of a quadratic equation: if D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are no real roots.

Front

How do you calculate the discriminant for the equation x² + 7x + 13?

Back

For the equation x² + 7x + 13, the discriminant is D = 7² - 4(1)(13) = 49 - 52 = -3.

Front

What is the significance of a negative discriminant?

Back

A negative discriminant indicates that the quadratic equation has no real solutions, meaning the graph does not intersect the x-axis.

Front

Solve for x: x² - 3x = 5.

Back

Rearranging gives x² - 3x - 5 = 0. Using the quadratic formula, x = (3 ± √(9 + 20)) / 2 = (3 ± √29) / 2.

Front

What are the roots of the equation 3x² = 16x + 12?

Back

Rearranging gives 3x² - 16x - 12 = 0. Using the quadratic formula, the roots are x = -0.67 and x = 6.

Front

If the discriminant is -2, how many real solutions are there?

Back

There are no real solutions because the discriminant is negative.

Front

What is the value of the discriminant for -2x² − x − 1 = 0?

Back

The discriminant is D = (-1)² - 4(-2)(-1) = 1 - 8 = -7.

Front

What does it mean if a quadratic equation has one real solution?

Back

It means the discriminant is zero (D = 0), indicating that the vertex of the parabola touches the x-axis.

Front

How can you determine the number of solutions from the discriminant?

Back

By evaluating the discriminant: D > 0 means 2 solutions, D = 0 means 1 solution, D < 0 means no real solutions.