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.

Tags

CCSS.HSA-REI.B.4B

Front

How do you complete the square for the equation x² + 2x - 10 = 0?

Back

1. Move -10 to the other side: x² + 2x = 10. 2. Take half of 2 (which is 1), square it (1² = 1), and add it to both sides: x² + 2x + 1 = 11. 3. Factor: (x + 1)² = 11. 4. Solve: x + 1 = ±√11, so x = -1 ± √11.

Tags

CCSS.HSA-REI.B.4B

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 discriminant in a quadratic equation?

Back

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature of the roots: if positive, two real roots; if zero, one real root; if negative, no real roots.

Tags

CCSS.HSA-REI.B.4B

Front

Solve the quadratic 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 (2² = 4), and add it to both sides: x² + 4x + 4 = 3. 3. Factor: (x + 2)² = 3. 4. Solve: x + 2 = ±√3, so x = -2 ± √3.

Tags

CCSS.HSA-REI.B.4B

Front

What is the axis of symmetry in a quadratic function?

Back

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, given by the formula x = -b/(2a) for the quadratic equation ax² + bx + c.

Front

Solve the quadratic 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 it to both sides: x² - 4x + 4 = 5. 3. Factor: (x - 2)² = 5. 4. Solve: x - 2 = ±√5, so x = 2 ± √5.

Tags

CCSS.HSA-REI.B.4B

Front

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

Back

The constant 'a' determines the direction of the parabola: if a > 0, the parabola opens upwards; if a < 0, it opens downwards.

Tags

CCSS.HSF-IF.C.7A

Front

Solve the quadratic equation x² + 12x + 10 = 0 by completing the square.

Back

1. Move 10 to the other side: x² + 12x = -10. 2. Take half of 12 (which is 6), square it (36), and add it to both sides: x² + 12x + 36 = 26. 3. Factor: (x + 6)² = 26. 4. Solve: x + 6 = ±√26, so x = -6 ± √26.

Tags

CCSS.HSA-REI.B.4B