Quadratic Equations

Algebra and Sequences

To solve, complete the square, isolate the squared bracket, then square root both signs
Writing ± at the square root stage is what produces the two solutions; forgetting it loses one of them
Finish by dealing with the number inside the bracket, which shifts both solutions equally
Leave the squared bracket intact once you have it, because multiplying it out again undoes the work
When there is a number in front of x², and only when the equation equals zero, you may divide every term by it first

Worked example

Solving and leaving the answer exact

Solve x² − 10x + 4 = 0, giving your answers in exact form.

Solution:

"Exact" rules out a decimal, so the answer will contain a surd and completing the square gives it directly
The coefficient of x² is already 1, so no factor needs taking out
Halve the coefficient of x: half of −10 is −5, so the bracket is (x − 5)²
(x − 5)² expands to x² − 10x + 25, which is 25 too many, so x² − 10x = (x − 5)² − 25
The equation becomes (x − 5)² − 25 + 4 = 0
Collect the constants: (x − 5)² − 21 = 0
Add 21 to both sides to isolate the square: (x − 5)² = 21
Take the square root of both sides, keeping both signs, since two numbers square to 21
x − 5 = ±√21
Add 5: x = 5 + √21 or x = 5 − √21
Leave the surd as it stands — writing 9.58 and 0.417 beside it risks losing the mark, because a decimal is not exact