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Home›Algebra›Quadratics

The quadratic formula

Al-Khwarizmi solved x² + 10x = 39 around 800 by completing a square. The quadratic formula is the same steps done once for every a, b and c.

A square drawn around 800

BackgroundMacTutor notes that the Babylonians, about 1800 BC, are often said to have solved quadratic equations, but they had no idea of an equation. What they had was a step-by-step recipe for problems we would write as quadratics, and the recipe is essentially completing the square, with answers that were always positive because they were usually lengths. Brahmagupta (598 to 665) gave an almost modern method that admitted negative quantities. Al-Khwarizmi, writing around 800 without zero or negatives, sorted quadratics into six types, gave a rule for each with a numerical example such as x² + 10x = 39, and proved each rule by completing a square geometrically, much like the picture below. In 1145 Abraham bar Hiyya's Liber embadorum became the first book published in Europe to give the complete solution.

Add 25 and x² + 10x becomes a square

Many quadratics do not factor with whole numbers. The method that always works is completing the square. Start with x² + 10x = 39. The left side is almost a perfect square: (x + 5)² = x² + 10x + 25, so it is missing 25. Add 25 to both sides to keep the equation balanced: x² + 10x + 25 = 64. Now the left side is (x + 5)², so (x + 5)² = 64. Take square roots: x + 5 = 8 or x + 5 = −8, which gives x = 3 or x = −13. The number to add is always the square of half the coefficient of x.

The geometric picture: a square of side x and two 5-by-x strips make x² + 10x. Adding the small 5-by-5 corner completes a square of side x + 5 with area 39 + 25 = 64.

The same steps with letters give the formula

Do the same steps with letters. Start from ax² + bx + c = 0, divide by a, move c/a across, add (b/(2a))² to both sides, and take square roots. The result is the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). The ± sign means two answers, one with plus and one with minus. To use it, first write the equation in standard form so you can read a, b and c with their signs. Watch b. In x² − 2x − 3 = 0, b = −2, so −b = 2.

Worked example: x² − 2x − 3 = 0 has a = 1, b = −2, c = −3. Then b² − 4ac = 4 + 12 = 16 and √16 = 4, so x = (2 ± 4)/2, which is 3 or −1. Factoring gives the same result, (x − 3)(x + 1) = 0. Now try 2x² + 3x − 1 = 0, which does not factor nicely: b² − 4ac = 9 + 8 = 17, so x = (−3 ± √17)/4. As decimals, x ≈ 0.281 or x ≈ −1.781. The formula hands you exact answers with a square root, and a calculator turns them into decimals when you need them.

Which method should you pick? If the equation has no x term, such as 5x² = 20, isolate x² and take square roots: x² = 4, x = ±2. If it factors quickly, factor. Otherwise use the formula, which never fails. In the lab, set a, b and c to any equation and the panel substitutes them into the formula step by step; the green dots on the x-axis are the solutions.

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y = 2x² + 3x − 1. Its x-intercepts are the solutions of 2x² + 3x − 1 = 0, about 0.281 and −1.781. Change a, b and c to solve other equations.
  1. What is the larger solution of x² − 2x − 3 = 0?
  2. In 3x² − 5x + 2 = 0, what is b?
  3. What is the positive solution of x² + 10x = 39?
  4. What is the larger solution of 2x² + 3x − 1 = 0, to two decimal places?
  5. What is the larger solution of x² + 4x + 1 = 0, to two decimal places?
  6. What number completes the square for x² + 6x?
  7. Solve x² − 6x + 9 = 0.
  8. What is the positive solution of 5x² = 20?
  9. In the quadratic formula, what is under the square root?
    • b² − 4ac
    • b² + 4ac
    • 4ac − b²
    • 2a
  10. What is the smaller solution of 2x² − 7x + 3 = 0? Write it as a fraction.
Quadratic formulax = (−b ± √(b² − 4ac)) / (2a)
Completing the square for x² + bxAdd (b/2)²
Square root propertyIf x² = k, then x = ±√k
Al-Khwarizmi's examplex² + 10x = 39, x = 3
Common mistakes
  • In x² − 4x − 5 = 0, putting −b = −4 into the formula and getting 1 and −5. b is −4, so −b is +4. Then b² − 4ac = 16 + 20 = 36 and x = (4 ± 6)/2, which is 5 or −1. Put 5 back in: 25 − 20 − 5 = 0.
  • Dividing only the square root by 2a: x = −b ± √(b² − 4ac)/(2a). The fraction bar runs under the whole top, −b included. For 2x² + 4x − 6 = 0, b² − 4ac = 16 + 48 = 64, so x = (−4 ± 8)/4 = 1 or −3. Dividing only the 8 gives −4 ± 2, that is −2 or −6, and neither satisfies the equation.
  • Solving (x + 2)² = 9 as x + 2 = 3, so x = 1 and nothing else. Both 3 and −3 square to 9. x + 2 = ±3 gives x = 1 or x = −5. Check the second one: (−5 + 2)² = (−3)² = 9.
Teacher's noteDerive the formula once from completing the square, in front of the class, and only then let students use it freely. A student who has seen the derivation knows why 2a sits under the whole numerator. After every formula answer, ask for a check by substitution. A dropped minus sign shows up straight away.
Key points
  • Completing the square: add (b/2)² to x² + bx.
  • Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a).
  • Read a, b and c with their signs from standard form.
  • No x term: use square roots. Easy factors: factor. Otherwise: the formula.
Sources for this course (6)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 중학교 1~3학년군 [9수02-19] 다항식의 곱셈과 인수분해(해설의 곱셈 공식 m(a + b), (a + b)², (a − b)², (a + b)(a − b), (x + a)(x + b), (ax + b)(cx + d)), [9수02-20] 이차방정식, [9수02-21] 이차함수의 개념, [9수02-22] 이차함수의 그래프(최댓값·최솟값은 x의 범위가 실수 전체인 경우만), 용어 인수·인수분해·완전제곱식·이차방정식·중근·근의 공식·이차함수·포물선·축·꼭짓점·최댓값·최솟값·평행이동, 「이차방정식은 해가 실수인 경우만」, 「근과 계수와의 관계는 다루지 않는다」, 「구한 해가 문제 상황에 적합한지 확인」; 고등학교 공통수학1 [10공수1-02-02] 실근과 허근·판별식, [10공수1-02-03] 근과 계수의 관계, [10공수1-02-06] 이차함수의 최대·최소와 실생활(제한된 범위, 공학 도구))
  • Textbook: OpenStax, College Algebra 2e , 5.1 Quadratic Functions (parabola, vertex, axis of symmetry x = −b/(2a), general form ax² + bx + c, standard form a(x − h)² + k also called vertex form, f(x) = 2x² + 4x − 4 = 2(x + 1)² − 6, f(x) = −3(x + 2)² + 4, vertex of 2x² − 6x + 7 by −b/(2a) (the completing-the-square example x² − 6x + 7 in the course is our own), satellite dish cross-sections, 80 ft fence: maximum area 800 sq ft at L = 20, newspaper revenue maximum at $31.80 and $2,528,100, ball H(t) = −16t² + 80t + 40: top at 2.5 s, 140 ft, ground after about 5.458 s, rock H(t) = −16t² + 96t + 112: openstax.org/books/college-algebra-2e/pages/5-1-quadratic-functions)
  • Textbook: OpenStax, Algebra and Trigonometry 2e, 2.5 Quadratic Equations (zero-product property, (x − 2)(x + 3) = x² + x − 6, square root property, completing the square, quadratic formula, discriminant table: openstax.org/books/algebra-and-trigonometry-2e/pages/2-5-quadratic-equations)
  • Reference value: NIST CODATA standard acceleration of gravity 9.80665 m/s², about 32.174 ft/s² (physics.nist.gov/cgi-bin/cuu/Value?gn)
  • Reference value: NIST, "Putting the Best Foot Forward" (1 foot = 0.3048 m exactly since 1959; nist.gov/pml/us-surveyfoot, checked 2026-10-19)
  • History block only: MacTutor, University of St Andrews, Quadratic, cubic and quartic equations (Babylonians about 1800 BC, algorithmic completing the square with positive answers; Euclid about 300 BC geometric; Brahmagupta (598-665) admits negative quantities; al-Khwarizmi (c 800) six types, x² + 10x = 39; Abraham bar Hiyya, Liber embadorum 1145: mathshistory.st-andrews.ac.uk/HistTopics/Quadratic_etc_equations).

Checked: 2026-10-18