Quadratic Formula

Front

What is the quadratic formula?

Back

The quadratic formula is given by x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a, b, and c are coefficients of the quadratic equation ax^2 + bx + c = 0.

Front

What does the discriminant of a quadratic equation indicate?

Back

The discriminant, given by b^2 - 4ac, indicates the nature of the roots of the quadratic equation. If it is positive, there are two distinct real roots; if zero, there is one real root; if negative, there are two complex roots.

Front

How do you derive the quadratic formula from a standard quadratic equation?

Back

To derive the quadratic formula, you can complete the square on the standard form of a quadratic equation ax^2 + bx + c = 0, leading to the formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Front

What is the vertex form of a quadratic equation?

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The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Front

How can you determine the vertex of a parabola given in standard form?

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The vertex (h, k) of a parabola given in standard form y = ax^2 + bx + c can be found using h = -\frac{b}{2a} and k = f(h).

Front

What is the axis of symmetry in a quadratic function?

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The axis of symmetry of a quadratic function is the vertical line x = -\frac{b}{2a}, which divides the parabola into two mirror-image halves.

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 (b^2 - 4ac). 3. Substitute a, b, and the discriminant into the quadratic formula. 4. Simplify to find the values of x.

Front

What is the significance of the coefficient 'a' in a quadratic equation?

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The coefficient 'a' determines the direction of the parabola (upward if a > 0, downward if a < 0) and affects the width of the parabola.

Front

How do you convert a quadratic equation from standard form to vertex form?

Back

To convert from standard form to vertex form, you can complete the square on the quadratic expression.

Front

What is the relationship between the roots of a quadratic equation and its factors?

Back

The roots of a quadratic equation ax^2 + bx + c = 0 correspond to the values of x that make the equation equal to zero, and can be expressed as factors: a(x - r1)(x - r2) where r1 and r2 are the roots.