To solve a quadratic equation that cannot be factored, we can use the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / (2a)
Where a, b, and c are the coefficients from the equation in the standard form ax² + bx + c = 0. Here’s how to apply the formula step by step:
- First, identify the values of a, b, and c from your equation.
- Next, calculate the discriminant, which is the part under the square root: D = b² – 4ac.
- If D > 0, there will be two distinct real solutions. If D = 0, there will be one real solution (a repeated root). If D < 0, the solutions will be complex numbers.
- Plug a, b, and the discriminant D into the quadratic formula to find the values of x.
For example, let’s solve the equation 2x² + 3x + 5 = 0. Here, a = 2, b = 3, and c = 5.
1. Calculate the discriminant: D = 3² – 4(2)(5) = 9 – 40 = -31.
2. Since D < 0, we know the solutions will be complex.
3. Now, substitute into the quadratic formula: x = (-3 ± √(-31)) / (2*2).
4. This simplifies to x = (-3 ± i√31) / 4, yielding two complex solutions.
Using the quadratic formula allows us to find the solutions for any quadratic equation, regardless of whether it can be factored easily or not.