The Quadratic Formula
Solve any quadratic with x = (-b ± √(b² - 4ac))/(2a), including ones that do not factorise, giving answers exactly or to a stated accuracy.
What a learner can do afterwards
- Identify a, b and c with their signs before substituting
- Give both roots, exact in surd form or rounded as the question asks
- Say why the formula was the right choice for a quadratic that does not factorise
1 · Read
Completing the square works on any quadratic, and doing it once on the general case, ax squared + bx + c = 0, produces the quadratic formula: x = (-b plus or minus the square root of b squared - 4ac) divided by 2a. You have carried out those moves before. The formula hands you the finished result as one line. When the pair hunt fails, this is the tool that always works.
The first move is the one that trips people up, so read a, b and c with their signs, as they sit in the equation. In 3x squared - 7x + 2, a is 3, b is -7 and c is 2. The minus belongs to b, not to the formula. When b is negative, -b in the formula turns positive, and that flip is where most slips hide. Keep the signs attached to the numbers until the substitution is done, and the formula does the rest.
Solve x squared - 6x - 3 = 0. Here a = 1, b = -6 and c = -3. Substituting gives x = (6 plus or minus the square root of 36 - 4(1)(-3)) divided by 2, which is (6 plus or minus the square root of 48) divided by 2. The number 48 is not a square, so the roots are not integers. Simplify the square root of 48 as 4 times the square root of 3, then split: x = (6 + 4 root 3) / 2 or (6 - 4 root 3) / 2, which divide cleanly to x = 3 + 2 root 3 or x = 3 - 2 root 3. Those surd forms are the exact answers.
Solve 2x squared - 3x - 2 = 0. Here a = 2, b = -3 and c = -2. Substituting: x = (3 plus or minus the square root of 9 - 4(2)(-2)) divided by 4, which is (3 plus or minus the square root of 25) divided by 4. This time the number under the root is a square: the square root of 25 is 5. So x = (3 + 5) / 4 or (3 - 5) / 4, giving x = 2 or x = -0.5. When the number under the root is a square, the formula just hands you the factorisation you could not find, with cleaner arithmetic.
When the question asks for decimal answers, the formula still runs the same way. Solve x squared - 4x - 1 = 0. Here a = 1, b = -4 and c = -1, so x = (4 plus or minus the square root of 16 - 4(1)(-1)) divided by 2, which is (4 plus or minus the square root of 20) divided by 2. The square root of 20 is about 4.472, so the two roots are (4 + 4.472) / 2 and (4 - 4.472) / 2, giving about 4.24 and about -0.24. Report both, to the accuracy the question asks for.
The number under the root, b squared - 4ac, is called the discriminant. If it is a square number, the quadratic also factorises. If it is positive but not a square, the roots are surds. If it is negative, no real number squares to it, so the equation has no real solutions. Before you substitute, try the pair method for a few seconds. If nothing appears, let the formula work.
Read a, b and c with their signs, substitute into x = (-b plus or minus the square root of b squared - 4ac) divided by 2a, and give both roots, exact in surd form or rounded as the question asks.
Readings 'Read a, b and c first', 'Checking your roots', 'Where the formula comes from' and 'What the discriminant tells you' are original to this dataset. All 6 extra questions are original to this dataset.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
30 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.