Solve Quadratic Using the Quadratic Formula: Step-by-Step Guide
The quadratic formula solves any quadratic equation of the form ax^2 + bx + c = 0. It comes from completing the square on the general equation. Before you finish, the discriminant, b^2 - 4ac, tells you how many real solutions there will be.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Write the equation in standard form. Move everything to one side so it reads ax^2 + bx + c = 0. Write down the values of a, b, and c.\[ax^2 + bx + c = 0\]
- Find the discriminant. Work out D = b^2 - 4ac. If D is positive, there are two real solutions. If D is 0, there is one solution. If D is negative, there are no real solutions.\[D = b^2 - 4ac\]
- Use the quadratic formula. Put a, b, and the discriminant into the formula: x = (-b +/- the square root of D) / (2a).\[x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
- Simplify and split into two answers. Simplify the square root if you can. Then work out one answer with the plus sign and one with the minus sign.\[x = \dfrac{-b + \sqrt{D}}{2a}, \quad x = \dfrac{-b - \sqrt{D}}{2a}\]
- Check each answer. Put each solution into the original equation. Both sides should be equal.\[a x_0^2 + b x_0 + c \stackrel{?}{=} 0\]
Worked example
Solve 2x^2 + 3x - 5 = 0 using the quadratic formula.
Step 1: The equation is already in standard form. a = 2, b = 3, and c = -5.
Step 2: The discriminant: D = 3^2 - 4(2)(-5) = 9 + 40 = 49. It is positive, so we expect two real solutions.
Step 3: Put it in the formula: x = (-3 +/- the square root of 49) / (2 x 2) = (-3 +/- 7) / 4.
Step 4: Use plus: x = (-3 + 7)/4 = 4/4 = 1. Use minus: x = (-3 - 7)/4 = -10/4 = -5/2.
Check: For x = 1: 2(1) + 3(1) - 5 = 0. For x = -5/2: 2(25/4) - 15/2 - 5 = 25/2 - 15/2 - 10/2 = 0. Both work.
The solutions are x = 1 and x = -5/2.
Tips and common mistakes
- Put the equation in the form ax^2 + bx + c = 0 first. The formula only works when the right side is 0.
- Be careful with signs when you pick out a, b, and c. In x^2 - 5x + 6 = 0, b is -5, not 5.
- Work out b^2 first, then 4ac, then subtract. Doing it in pieces avoids sign mistakes.
- If the discriminant is a perfect square (like 49), the answers are fractions or whole numbers, and the equation could have been factored.
- If the discriminant is negative, there are no real solutions. In later classes you will learn about complex numbers.
- Simplify the square root before you split into two answers. For example, the square root of 50 is 5 times the square root of 2.
- The 2a on the bottom divides everything on top, including -b.
- Quick check: the two solutions should add up to -b/a and multiply to c/a.