Solve Quadratic Equation: Step-by-Step Guide
A quadratic equation has an x^2 term and looks like ax^2 + bx + c = 0. It can have up to two solutions. There are several ways to solve one: factoring, completing the square, taking square roots, or using the quadratic formula. When the trinomial factors nicely, factoring is usually the fastest.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Make one side equal 0. Move all the terms to the left side so the right side is 0. It should look like ax^2 + bx + c = 0.\[ax^2 + bx + c = 0\]
- Factor the trinomial. Find two numbers that multiply to a x c and add to b. Use them to write the trinomial as a product of two binomials.\[ax^2 + bx + c = (px + r)(qx + s)\]
- Use the zero product property. If two things multiply to 0, then at least one of them must be 0. Set each factor equal to 0.\[(px + r)(qx + s) = 0 \;\Rightarrow\; px + r = 0 \;\text{or}\; qx + s = 0\]
- Solve each small equation. Solve each of the equations for x. These are your solutions.\[x = -\tfrac{r}{p} \quad \text{or} \quad x = -\tfrac{s}{q}\]
- Check your answers. Put each solution back into the original equation. Both sides should be 0 (or equal).\[a(x_0)^2 + b(x_0) + c \stackrel{?}{=} 0\]
Worked example
Solve x^2 - 5x + 6 = 0 by factoring.
Step 1: The equation x^2 - 5x + 6 = 0 already has 0 on the right side. Here a = 1, b = -5, and c = 6.
Step 2: Find two numbers that multiply to 6 and add to -5. They are -2 and -3.
Step 3: Factor: x^2 - 5x + 6 = (x - 2)(x - 3) = 0.
Step 4: Set each factor equal to 0: x - 2 = 0 or x - 3 = 0.
Step 5: Solve each one: x = 2 or x = 3.
Check: For x = 2: 4 - 10 + 6 = 0. For x = 3: 9 - 15 + 6 = 0. Both work.
Tips and common mistakes
- Always get 0 on one side before you factor. The zero product property only works when the product equals 0.
- When a = 1, look for two numbers that multiply to c and add to b.
- When a is not 1, use the AC method: split the middle term, then factor by grouping.
- Watch your signs. If c is positive and b is negative, both numbers are negative.
- If the trinomial will not factor, use the quadratic formula or completing the square.
- Learn the special patterns: a^2 - b^2 = (a - b)(a + b), and a^2 + 2ab + b^2 = (a + b)^2.
- Check for a common factor first. It makes the factoring much easier.
- Always check your answers by putting them back in.