AI Tutor
Imagine writing code for a rocket simulator: your quadratic trajectory formula predicts the booster lands at t=4t = 4 seconds. If that number is even slightly wrong, the rocket crashes into the ground instead of touching down safely.
How do you guarantee a solution is 100% correct before running the simulation? You verify it by plugging it back into the original equation.

The Meaning of a Solution

A quadratic equation is any equation that can be written in the form ax2+bx+c=0ax^2 + bx + c = 0, where a≠0a \neq 0. A solution (or root) is simply an input value for the variable that makes the left side and the right side mathematically equal.
What does this verification look like step by step when dealing with tricky negative signs?

Verifying Integer and Fractional Roots

Let's check whether x=3x = 3 and x=βˆ’12x = -\frac{1}{2} are genuine solutions to 2x2βˆ’5xβˆ’3=02x^2 - 5x - 3 = 0.
Substitute x=3x = 3 into every instance of the variable: 2(3)2βˆ’5(3)βˆ’3=2(9)βˆ’15βˆ’3=18βˆ’18=02(3)^2 - 5(3) - 3 = 2(9) - 15 - 3 = 18 - 18 = 0. Since 0=00 = 0, x=3x = 3 is verified.
Now test x=βˆ’12x = -\frac{1}{2}: 2(βˆ’12)2βˆ’5(βˆ’12)βˆ’3=2(14)+52βˆ’3=12+52βˆ’3=3βˆ’3=02\left(-\frac{1}{2}\right)^2 - 5\left(-\frac{1}{2}\right) - 3 = 2\left(\frac{1}{4}\right) + \frac{5}{2} - 3 = \frac{1}{2} + \frac{5}{2} - 3 = 3 - 3 = 0. Both solutions are valid.
The most common algebra error when checking negative solutions is dropping parentheses: (βˆ’3)2(-3)^2 equals +9+9, but βˆ’32-3^2 means βˆ’(32)-(3^2), which is βˆ’9-9.
Always wrap negative numbers in parentheses before applying exponents to ensure the negative sign is squared as well.