How to Solve Quadratics Without A Calculator

The Quadratic Formula

The quadratic formula is the most straightforward method for solving quadratic equations. It works for any quadratic equation in the standard form:

The formula to find the roots (solutions) is:

Steps to Use the Quadratic Formula

1. Identify the coefficients a, b, and c from the equation.
2. Calculate the discriminant (b² - 4ac).
3. Take the square root of the discriminant.
4. Add and subtract this value from -b.
5. Divide each result by 2a to get the two solutions.

Completing the Square

Completing the square transforms a quadratic equation into a perfect square trinomial, making it easier to solve. This method is particularly useful when the equation is not easily factorable.

Steps to Complete the Square

1. Start with the equation in the form ax² + bx + c = 0.
2. Divide all terms by a if a ≠ 1.
3. Move the constant term to the other side.
4. Take half of the coefficient of x, square it, and add it to both sides.
5. Write the left side as a perfect square and solve for x.

Example: Solve x² + 6x + 8 = 0

1. Divide by 1 (no change).
2. Move 8 to the other side: x² + 6x = -8.
3. Half of 6 is 3, squared is 9. Add 9 to both sides: x² + 6x + 9 = 1.
4. Write as (x + 3)² = 1.
5. Take square roots: x + 3 = ±1 → x = -3 ± 1 → x = -2 or x = -4.

Factoring Quadratics

Factoring is the most efficient method when the quadratic can be easily expressed as a product of two binomials. It works best for equations where a, b, and c are small integers.

Steps to Factor a Quadratic

1. Find two numbers that multiply to a × c and add to b.
2. Rewrite the middle term using these numbers.
3. Factor by grouping.
4. Set each factor equal to zero and solve for x.

Example: Solve x² + 5x + 6 = 0

1. Find numbers that multiply to 6 and add to 5: 2 and 3.
2. Rewrite: x² + 2x + 3x + 6 = 0.
3. Factor: (x² + 2x) + (3x + 6) = x(x + 2) + 3(x + 2) = (x + 2)(x + 3).
4. Set each factor to zero: x + 2 = 0 → x = -2 or x + 3 = 0 → x = -3.

Example Problems

Problem 1: Using the Quadratic Formula

Solve 2x² - 4x - 6 = 0

1. Identify a=2, b=-4, c=-6.
2. Discriminant: (-4)² - 4(2)(-6) = 16 + 48 = 64.
3. √64 = 8.
4. -b ± 8 = 4 ± 8 → 12 or -4.
5. Divide by 2a: 12/4 = 3 or -4/4 = -1.

Solutions: x = 3 and x = -1.

Problem 2: Completing the Square

Solve x² - 6x + 5 = 0

1. Divide by 1 (no change).
2. Move 5 to the other side: x² - 6x = -5.
3. Half of -6 is -3, squared is 9. Add 9 to both sides: x² - 6x + 9 = 4.
4. Write as (x - 3)² = 4.
5. Take square roots: x - 3 = ±2 → x = 3 ± 2 → x = 5 or x = 1.

Solutions: x = 5 and x = 1.

Problem 3: Factoring

Solve x² - 7x + 10 = 0

1. Find numbers that multiply to 10 and add to -7: -2 and -5.
2. Rewrite: x² - 2x - 5x + 10 = 0.
3. Factor: (x² - 2x) + (-5x + 10) = x(x - 2) - 5(x - 2) = (x - 2)(x - 5).
4. Set each factor to zero: x - 2 = 0 → x = 2 or x - 5 = 0 → x = 5.

Solutions: x = 2 and x = 5.