An equation with a square
A quadratic equation is an equation whose highest power of the variable is two. Its standard form is ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
For example, x² − 5x + 6 = 0 is quadratic, with a = 1, b = −5, and c = 6. A root is a value of x that makes the equation true.
You can picture the expression ax² + bx + c as a parabola. Real roots are the x-coordinates where this graph touches or crosses the x-axis.
The coefficient of x² must be non-zero.
Start with factorisation
For x² − 5x + 6, look for two numbers whose product is 6 and whose sum is −5. Those numbers are −2 and −3.
So x² − 5x + 6 = (x − 2)(x − 3). If a product is zero, at least one factor must be zero. Therefore x = 2 or x = 3.
Check each answer by substituting it into the original equation. For x = 2: 4 − 10 + 6 = 0.
The quadratic formula
When factorisation is awkward, use x = (−b ± √(b² − 4ac)) / (2a). Identify a, b, and c including their signs before substituting.
The expression b² − 4ac is called the discriminant. If it is positive, there are two distinct real roots. If it is zero, there is one repeated real root. If it is negative, there are no real roots.
A negative discriminant means no real roots; complex roots are a later topic.
Try it yourself
Solve x² − 7x + 12 = 0. The numbers −3 and −4 multiply to 12 and add to −7. The factors are (x − 3)(x − 4), so the roots are 3 and 4.
Now explain why x² + 1 = 0 has no real root. Any real number squared is non-negative, so adding 1 cannot give zero.