Solving Quadratics by using the Quadratic Formula

Front

What does the discriminant (b² - 4ac) indicate about the solutions of a quadratic equation?

Back

If the discriminant is positive, the quadratic equation has two distinct real solutions. If it is zero, there is one real solution (a repeated root). If it is negative, there are no real solutions.

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

How do you apply the Quadratic Formula to the equation x² + 4x + 3 = 0?

Back

Identify a = 1, b = 4, c = 3. Calculate the discriminant: b² - 4ac = 4² - 4(1)(3) = 16 - 12 = 4 (positive). Then, apply the formula: x = (-4 ± √4) / (2*1) = (-4 ± 2) / 2, giving x = -1 and x = -3.

Front

What are the steps to solve a quadratic equation using the Quadratic Formula?

Back

1. Identify coefficients a, b, and c from the equation. 2. Calculate the discriminant (b² - 4ac). 3. Determine the number of solutions based on the discriminant. 4. Apply the Quadratic Formula to find the solutions.

Front

What does it mean if the discriminant is zero?

Back

If the discriminant is zero, the quadratic equation has exactly one real solution, also known as a repeated or double root.

Front

What is the significance of a negative discriminant in a quadratic equation?

Back

A negative discriminant indicates that the quadratic equation has no real solutions, but two complex solutions.

Front

How can you rearrange the equation x² + 4x - 40 = -8 to use the Quadratic Formula?

Back

Rearrange to standard form: x² + 4x + 8 - 40 = 0, which simplifies to x² + 4x + 32 = 0.

Front

What are the solutions to the equation 2x² + 7x - 15 = 0 using the Quadratic Formula?

Back

Using the formula, the solutions are x = -5 and x = 1.5.

Front

What is the role of 'a' in the Quadratic Formula?

Back

In the Quadratic Formula, 'a' is the coefficient of x² and determines the shape and direction of the parabola represented by the quadratic equation.

Front

What does it mean for a quadratic equation to have two real solutions?

Back

It means that the graph of the quadratic function intersects the x-axis at two distinct points.