Completing the Square

Front

What is 'completing the square'?

Back

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form (x + p)² = q, which makes it easier to find the roots.

Front

What is the standard form of a quadratic equation?

Back

The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

How do you complete the square for the expression x² + 6x?

Back

To complete the square, take half of the coefficient of x (which is 6), square it (3² = 9), and add it: x² + 6x + 9 = (x + 3)².

Front

What is the vertex form of a quadratic equation?

Back

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

Front

What is the significance of the vertex in a quadratic function?

Back

The vertex represents the maximum or minimum point of the parabola, depending on the direction it opens.

Front

How do you find the roots of a quadratic equation using completing the square?

Back

After rewriting the equation in vertex form, set it equal to zero and solve for x to find the roots.

Front

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

Back

The sum of the roots is -b/a and the product of the roots is c/a, according to Vieta's formulas.

Front

What is the discriminant of a quadratic equation?

Back

The discriminant is given by b² - 4ac and determines the nature of the roots: if positive, two real roots; if zero, one real root; if negative, no real roots.

Front

How can you determine if a quadratic opens upwards or downwards?

Back

A quadratic opens upwards if the coefficient a > 0 and downwards if a < 0.

Front

What is the axis of symmetry in a quadratic function?

Back

The axis of symmetry is the vertical line x = -b/(2a) that divides the parabola into two mirror-image halves.