Quadratic Equations

What is a quadratic equation?

A quadratic equation is an equation that can be written in standard form

ax² + bx + c = 0, where a ≠ 0.

The numbers a, b and c are the coefficients. The highest power of x is 2: that is what “quadratic” means (it comes from the Latin word for a square). If a were 0 there would be no x² term and the equation would be linear.

A solution (or root) is a value of x that makes the equation true. A quadratic equation has two real solutions, one, or none, and the graph below shows why.

Your turn

Write 2x² = 3x + 5 in standard form and name a, b and c.

The graph: a parabola

The graph of y = ax² + bx + c is a U-shaped curve called a parabola. The picture at the top of the page shows y = x² − 2x − 3. Four features matter most:

Solving by square roots

When the equation has an x² (or a squared bracket) and no other x term, get the square by itself and take the square root of both sides. Both 4² and (−4)² equal 16, so there are two square roots: write ±, read “plus or minus”.

If the number isn’t a perfect square, leave a square root in the answer: x² = 7 gives x = ±√7, about ±2.65.

Solving by factoring

Factoring rests on one fact, the zero-product property: if AB = 0, then A = 0 or B = 0. If two numbers multiply to 0, one of them has to be 0.

1. Move every term to one side so the other side is 0.
2. Factor the quadratic into two brackets.
3. Set each bracket equal to 0 and solve.
4. Check each answer in the original equation.

When a = 1

To factor x² + bx + c, find two numbers that multiply to c and add to b.

When a ≠ 1: the ac method

For ax² + bx + c, find two numbers that multiply to a · c and add to b. Use them to split the middle term into two, then factor in pairs (this is called factoring by grouping).

Completing the square

Not every quadratic factors nicely. Completing the square works on any of them: rewrite the left side as a perfect square (x + h)², then use square roots. The key fact is that x² + 2hx + h² = (x + h)², so you add the square of half the x-coefficient.

1. If a ≠ 1, divide every term by a.
2. Move the constant to the right side.
3. Take half of the x-coefficient, square it, and add it to both sides.
4. Write the left side as a square and take square roots.

The quadratic formula

Completing the square on ax² + bx + c = 0 with letters instead of numbers gives a formula that solves every quadratic equation:

x = (−b ± √(b² − 4ac))/(2a)

Write the equation in standard form first, read off a, b and c with their signs, and put them in.

The discriminant: how many solutions?

The part under the square root, b² − 4ac, is called the discriminant. You can tell how many real solutions there are before you solve:

Vertex form

A quadratic written as y = a(x − h)² + k is in vertex form. Its vertex is (h, k) and its axis of symmetry is x = h, so you can read them straight off. Watch the sign: (x − 2)² means h = 2, and (x + 2)² means h = −2.

Word problems

Quadratics model areas (length times width) and objects thrown or dropped. Write the equation, get 0 on one side, solve, and then ask whether each answer makes sense: a length or a time can’t be negative.

The question in Example 2: a ball is thrown upward at 20 m/s from the top of a 25 m cliff. In the model, −5t² is the pull of gravity: gravity is about 9.8 meters per second squared, and the 5 is half of that, rounded; 20t is the throw and 25 is the starting height. When does it hit the ground?

Common mistakes

Practice problems

Printable worksheet (PDF)

Solve each one on paper (problems 1 to 8 ask you to solve for x), then open the answer to check your work.

1. x² = 49
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   x = ±7Take square rootsx = −7 or x = 7
2. 3x² − 75 = 0
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   3x² = 75Add 75x² = 25Divide by 3x = ±5Take square rootsx = −5 or x = 5
3. (x − 2)² = 9
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   x − 2 = ±3Take square rootsx = 2 ± 3Add 2x = 5 or x = −1
4. x² + 7x + 12 = 0
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   3 · 4 = 12 and 3 + 4 = 7Multiply to 12, add to 7(x + 3)(x + 4) = 0Factorx = −4 or x = −3Zero-product property
5. x² − 9x = 0
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   x(x − 9) = 0Factor out xx = 0 or x = 9Zero-product property
6. 3x² − 10x − 8 = 0
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   3 · (−8) = −24a · c−12 · 2 = −24 and −12 + 2 = −10Multiply to −24, add to −103x² − 12x + 2x − 8 = 0Split the middle term3x(x − 4) + 2(x − 4) = 0Factor each pair(3x + 2)(x − 4) = 0Factor out (x − 4)x = −2/3 or x = 4Zero-product property
7. Solve by completing the square: x² + 8x + 11 = 0
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   x² + 8x = −11Subtract 11x² + 8x + 16 = −11 + 16Half of 8 is 4, and 4² = 16(x + 4)² = 5Perfect squarex + 4 = ±√5Take square rootsx = −4 ± √5About −1.76 and −6.24
8. Use the quadratic formula: 2x² + x − 6 = 0
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   a = 2, b = 1, c = −6Read the coefficients1² − 4(2)(−6) = 49b² − 4acx = (−1 ± 7)/4√49 = 7x = 3/2 or x = −26/4 = 3/2 and −8/4 = −2
9. How many real solutions does 2x² − x + 3 = 0 have?
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   (−1)² − 4(2)(3) = −23The discriminantNoneIt is negative, so there is no real solution
10. Write y = x² + 4x + 1 in vertex form and give the vertex.
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    y = (x² + 4x + 4) − 4 + 1Add and subtract 2² = 4y = (x + 2)² − 3Vertex formVertex (−2, −3)h = −2 and k = −3
11. A ball’s height is h = −5t² + 10t + 15 meters after t seconds. When does it hit the ground?
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    −5t² + 10t + 15 = 0Set h = 0t² − 2t − 3 = 0Divide by −5(t − 3)(t + 1) = 0Factort = 3t = −1 makes no sense here: it lands after 3 seconds

History: from Babylonian tablets to al-Khwarizmi

Problems that we would now write as quadratic equations appear on clay tablets from Babylonia (in present-day Iraq) from around 1800 BC. The scribes had no symbols and no idea of an “equation” as we use the word. Instead they wrote out a recipe of steps for a problem about lengths and areas, and historians describe their method as essentially completing the square. Their answers were positive numbers, which makes sense when the answer is a length.

More than two thousand years later, in the early 800s, the scholar Muhammad ibn Musa al-Khwarizmi worked in Baghdad at the House of Wisdom. His book on al-jabr and al-muqabala (“restoring” and “balancing”) gives us the word algebra. He sorted equations into six types, wrote everything in words with no symbols and no negative numbers, and backed up his rules with geometric pictures.

One of his examples, in words, was “a square and ten roots equal thirty-nine”: x² + 10x = 39. His recipe was to halve the 10 to get 5, square it to get 25, add the 39 to get 64, take the square root to get 8, and subtract the 5 to get 3. For the proof he drew a square of side x with four rectangles of breadth 10/4 on its sides, then filled in the four small corner squares, each with area 25/4. That adds 25 in total and makes one big square of area 39 + 25 = 64, so its side, x + 5, is 8.

Sources

• J. J. O’Connor and E. F. Robertson, MacTutor History of Mathematics, University of St Andrews: Quadratic, cubic and quartic equations (Babylonian problems from about 1800 BC, solved by a method that is essentially completing the square, with no notion of an equation and positive answers; al-Khwarizmi’s six types of equation and geometric proofs).
• J. J. O’Connor and E. F. Robertson, MacTutor: Abu Ja’far Muhammad ibn Musa Al-Khwarizmi (the House of Wisdom in Baghdad; the word “algebra” from al-jabr; the example “a square and ten roots equal thirty-nine” and its geometric proof).
• Common Core State Standards for Mathematics: Reasoning with Equations and Inequalities (A-REI.B.4, solving quadratic equations), Seeing Structure in Expressions (A-SSE.B.3, factoring and completing the square), Interpreting Functions (F-IF.C.7, graphing quadratics) and Creating Equations (A-CED.A.1), which this lesson supports.

Credits: all graphs on this page are original SVG drawings made for Jugami. Every equation, solution and step was checked by a script.