Solving Quadratic Equations

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 (b² - 4ac) tell us about a quadratic equation?

Back

The discriminant indicates the nature of the roots of the quadratic equation: if it's positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); if it's negative, there are no real roots (the roots are complex).

Front

Solve for x using the Quadratic Formula: 2x² + 6x - 7 = 0

Back

x = (-6 ± √(6² - 4*2*(-7))) / (2*2) = (-6 ± √(36 + 56)) / 4 = (-6 ± √92) / 4 = (-6 ± 2√23) / 4 = -3/2 ± √23/2.

Front

Solve for x using the Quadratic Formula: x² + 4x - 32 = 0

Back

x = (-4 ± √(4² - 4*1*(-32))) / (2*1) = (-4 ± √(16 + 128)) / 2 = (-4 ± √144) / 2 = (-4 ± 12) / 2 = 4 or -8.

Front

What is the Zero Product Property?

Back

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. This is used to solve quadratic equations that can be factored.

Front

Solve the quadratic equation by factoring: x² - 7x + 10 = 0

Back

Factoring gives (x - 2)(x - 5) = 0, so x = 2 or x = 5.

Front

What is a quadratic equation?

Back

A quadratic equation is a polynomial equation of degree 2, typically in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

What are the coefficients in the quadratic equation 3x² + 2x - 5 = 0?

Back

The coefficients are: a = 3, b = 2, c = -5.

Front

How can you determine if a quadratic equation can be factored easily?

Back

If the discriminant (b² - 4ac) is a perfect square, the quadratic can often be factored easily.

Front

What is the vertex of a parabola represented by a quadratic equation?

Back

The vertex of a parabola given by the equation y = ax² + bx + c can be found using the formula (-b/(2a), f(-b/(2a))).