Solving Quadratics by Completing the Square

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

What does it mean to complete 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, making it easier to solve for x.

Front

How do you complete the square for the equation x² + 6x - 3 = 0?

Back

1. Move -3 to the other side: x² + 6x = 3. 2. Take half of 6 (which is 3), square it (9), and add to both sides: x² + 6x + 9 = 12. 3. Factor: (x + 3)² = 12. 4. Solve: x + 3 = ±√12, so x = -3 ± √12.

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 quadratic formula?

Back

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the roots of a quadratic equation.

Front

How do you find the roots of the equation x² - 4x - 1 = 0 by completing the square?

Back

1. Move -1 to the other side: x² - 4x = 1. 2. Take half of -4 (which is -2), square it (4), and add to both sides: x² - 4x + 4 = 5. 3. Factor: (x - 2)² = 5. 4. Solve: x - 2 = ±√5, so x = 2 ± √5.

Front

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

Back

The discriminant (b² - 4ac) determines the nature of the roots: if > 0, two real roots; if = 0, one real root; if < 0, no real roots.

Front

How do you solve the equation (2x - 1)² = 49?

Back

1. Take the square root of both sides: 2x - 1 = ±7. 2. Solve for x: 2x = 8 or 2x = -6, so x = 4 or x = -3.

Front

What is the axis of symmetry for a quadratic equation in standard form?

Back

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

Front

How do you solve x² + 10x = -9 by completing the square?

Back

1. Move -9 to the other side: x² + 10x + 9 = 0. 2. Take half of 10 (which is 5), square it (25), and add to both sides: x² + 10x + 25 = 16. 3. Factor: (x + 5)² = 16. 4. Solve: x + 5 = ±4, so x = -1 or x = -9.