📖Definition
The quadratic formula is a mathematical formula that provides the solution(s) to a quadratic equation of the form ax² + bx + c = 0. It gives you the values of x that make the equation true.
📐Formula
Here a is the coefficient of x², b is the coefficient of x, and c is the constant term. The ± symbol means you work the sum out twice: once adding the square root, once subtracting it. That gives the two solutions.
🔎What the discriminant tells you
Two real solutions
If b² − 4ac is positive, the square root is real and the equation has two distinct real solutions. The parabola crosses the x-axis twice.
One real solution
If b² − 4ac equals zero, both formula results are the same, giving one repeated solution. The parabola just touches the x-axis.
No real solutions
If b² − 4ac is negative, the square root is not real, so there are no real solutions. The parabola never reaches the x-axis.
📝Step-by-Step Guide
Identify the coefficients
Write your equation in standard form (ax² + bx + c = 0) and identify the values of a, b, and c.
Calculate the discriminant
The discriminant tells you how many solutions exist. Calculate b² - 4ac.
Apply the formula
Substitute your values into the quadratic formula and simplify.
Calculate both solutions
Use the + sign for one solution and the - sign for the other.
🧮Worked Examples
Example 1: answers that are surds
Solve x² − 4x − 1 = 0, leaving your answer in exact form.
- 1Read off a, b and c. The equation is already in standard form, so a = 1, b = −4 and c = −1.
- 2Find the discriminant. b² − 4ac = (−4)² − 4(1)(−1) = 16 + 4 = 20. It is positive, so expect two real solutions.
- 3Substitute. x = (4 ± √20) / 2. Note that −b is −(−4), which is +4.
- 4Simplify the surd. √20 = √(4 × 5) = 2√5, so x = (4 ± 2√5) / 2.
- 5Divide every term by 2. Both 4 and 2√5 are divided by the 2 underneath, giving x = 2 ± √5.
Answer: x = 2 + √5 or x = 2 − √5 (about 4.24 and −0.24)
The discriminant was not a perfect square, so the answers cannot be whole numbers or simple fractions. Factoring would never have worked here.
Example 2: no real solutions
Solve 2x² + 3x + 5 = 0.
- 1Read off a, b and c. a = 2, b = 3 and c = 5.
- 2Find the discriminant. b² − 4ac = 3² − 4(2)(5) = 9 − 40 = −31.
- 3Interpret the sign. The discriminant is negative. You cannot take the square root of a negative number and get a real result.
- 4State the conclusion. There are no real solutions. Sketching y = 2x² + 3x + 5 confirms it: the curve sits entirely above the x-axis.
Answer: No real solutions
At A-Level, IB Higher Level or US Algebra 2 you continue with imaginary numbers and get x = (−3 ± i√31) / 4. Before that point, "no real solutions" is the expected answer.
Example 3: a word problem about area
A rectangular vegetable bed is 3 feet longer than it is wide. Its area is 40 square feet. Find its dimensions.
- 1Name the unknown. Let the width be w feet. The length is then w + 3 feet.
- 2Write an equation. Area is length times width, so w(w + 3) = 40.
- 3Rearrange to standard form. Expand to get w² + 3w = 40, then subtract 40 from both sides: w² + 3w − 40 = 0.
- 4Apply the formula. a = 1, b = 3, c = −40. The discriminant is 9 + 160 = 169, and √169 = 13. So w = (−3 ± 13) / 2.
- 5Reject the impossible root. The two results are w = 5 and w = −8. A width cannot be negative, so w = 5.
Answer: The bed is 5 ft wide and 8 ft long. Check: 5 × 8 = 40 ✓
Word problems almost always produce one root that has to be discarded. Always ask whether a negative length, time or price makes sense.
Example 4: a word problem about projectile height
A ball is kicked from ground level. Its height in metres after t seconds is h = 20t − 5t². At what times is the ball 15 m above the ground?
- 1Set up the equation. Put h = 15, so 20t − 5t² = 15.
- 2Rearrange to standard form. Move everything to one side: −5t² + 20t − 15 = 0.
- 3Simplify first. Divide every term by −5 to get t² − 4t + 3 = 0. Smaller numbers mean fewer slips.
- 4Apply the formula. a = 1, b = −4, c = 3. The discriminant is 16 − 12 = 4, and √4 = 2. So t = (4 ± 2) / 2.
- 5Interpret both answers. The results are t = 3 and t = 1. Both are valid here.
Answer: t = 1 second on the way up, and t = 3 seconds on the way down
Here both roots have meaning. The ball passes 15 m twice, once rising and once falling. Dividing through by a common factor before you start is a habit worth building.
🧭Standard form and other notations
Every equation must be in standard form before you start. Standard form is ax² + bx + c = 0.
Many calculators and textbooks drop the superscript and write ax2 + bx + c = 0. Both mean the same thing.
Rearrange first if the equation looks different. For example, 3x² = 2x + 8 becomes 3x² − 2x − 8 = 0.
Now read the three numbers straight off: a = 3, b = −2, c = −8. Keep the minus signs attached.
🧭Three ways of solving quadratic equations
There are three standard methods. You can factor, complete the square, or substitute into the formula.
Factoring is fastest when the numbers are friendly. x² − 5x + 6 = 0 factors to (x − 2)(x − 3) = 0, so x = 2 or x = 3.
Completing the square works on any quadratic. It takes longer, but it shows where the turning point sits.
The formula also works on any quadratic, and it needs no guesswork. That is why most students reach for it first.
🧭Completing the square, step by step
Start with x² + 6x + 5 = 0. Move the constant across to get x² + 6x = −5.
Take half of the x coefficient and square it. Half of 6 is 3, and 3² is 9.
Add 9 to both sides: x² + 6x + 9 = 4. The left side is now (x + 3)².
Take the square root of both sides: x + 3 = ±2. So x = −1 or x = −5.
Completing the square on the general equation ax² + bx + c = 0 is exactly how the formula is derived.
🧭The p-q formula used in Germany and the Nordics
Students in Germany, Austria, Sweden and several neighbouring countries usually learn a slightly different version. It is called the p-q formula, or pq-formeln.
It applies to equations written as x² + px + q = 0, where the x² term has a coefficient of 1. The formula is x = −p/2 ± √((p/2)² − q).
Try it on x² + 6x + 5 = 0. Here p = 6 and q = 5, so x = −3 ± √(9 − 5) = −3 ± 2. That gives x = −1 or x = −5, exactly as before.
If the x² term has a coefficient, divide every term by it first. For 2x² + 12x + 10 = 0, divide by 2 to get x² + 6x + 5 = 0.
The two formulas are the same rule written differently. Substituting a = 1, b = p and c = q into the standard version produces the p-q formula.
🧭What the solutions look like on a graph
Every quadratic draws a parabola, a smooth U-shaped curve. If a is positive the curve opens upwards, and if a is negative it opens downwards.
The solutions of ax² + bx + c = 0 are the x-values where the curve meets the x-axis. They are also called the roots, or the x-intercepts.
That is why the discriminant controls the number of solutions. Two roots means the curve cuts the axis twice, one root means it just touches, and no real roots means it misses entirely.
The turning point sits exactly halfway between the two roots, on the line x = −b/2a. Notice that this is the formula with the square root removed.
🧭Sum and product of the roots
There is a shortcut that lets you check answers without redoing the whole calculation.
For ax² + bx + c = 0, the two roots always add up to −b/a, and they always multiply to c/a.
Take x² − 5x + 6 = 0, whose roots are 2 and 3. The sum is 5, which matches −(−5)/1. The product is 6, which matches 6/1.
This is a fast way to catch a sign error. If your two answers do not add to −b/a, something went wrong.
The result is known as Vieta's formulas and appears in A-Level, IB Higher Level and US Algebra 2 courses.
🧭Where quadratics show up in real life
Anything thrown, kicked or dropped follows a parabolic path, so height against time is a quadratic. Working out when a ball lands means solving one.
Area problems produce quadratics whenever two dimensions are linked, such as a garden that is 3 m longer than it is wide.
Businesses use them for revenue. If raising the price lowers the number of items sold, total revenue is a quadratic, and its highest point gives the best price.
Braking distance grows with the square of speed, which is why stopping distance tables in driving manuals are quadratic rather than linear.
Satellite dishes, car headlights and solar cookers are parabolic because that shape focuses everything arriving in parallel onto a single point.
🧭Checking your answers
Always substitute each answer back into the original equation.
Test x = 2 in x² − 5x + 6 = 0. That gives 4 − 10 + 6, which equals 0, so the answer is correct.
A quick sketch is a second check. Your solutions are the points where the parabola crosses the x-axis.
For word problems, check the answer makes sense as well as being arithmetically right. Negative lengths and negative times usually have to be discarded.
⚠️Common Mistakes to Avoid
- Forgetting to include the negative sign in front of b
- Making errors when calculating the discriminant (b² - 4ac)
- Dividing only part of the numerator by 2a instead of the entire expression
- Not simplifying the square root when possible
- Forgetting that ± gives two different answers
✏️Practice Problems
Solve x² - 5x + 6 = 0
💡 Hint: Identify a=1, b=-5, c=6
Answer: x = 2 or x = 3
Solve 2x² + 7x - 15 = 0
💡 Hint: Remember to use a=2 when dividing
Answer: x = 1.5 or x = -5
Solve 3x² - 2x - 8 = 0
Answer: x = 2 or x = -4/3
Solve x² + 4x + 4 = 0
Answer: x = -2
Identify a, b, and c in 2x² - 3x + 1 = 0
Answer: a = 2, b = -3, c = 1
Solve x² - 2x - 8 = 0 using the quadratic formula
Answer: x = 4 or x = -2
Find the discriminant of 2x² + 4x + 5 = 0
Answer: Δ = -24
Solve 4x² + 4x - 3 = 0
Answer: x = 1/2 or x = -3/2
For what value of k does x² + 6x + k = 0 have one real solution?
Answer: k = 9
Write 5x² = 3x − 2 in standard form and state a, b and c.
Answer: 5x² − 3x + 2 = 0, so a = 5, b = −3, c = 2
Solve x² − 6x + 7 = 0 and leave your answer in surd form.
💡 Hint: The discriminant is 8, and √8 simplifies to 2√2.
Answer: x = 3 + √2 or x = 3 − √2
A rectangle is 2 cm longer than it is wide. Its area is 24 cm². Find the width.
💡 Hint: Let the width be w. Then w(w + 2) = 24.
Answer: Width = 4 cm (the negative root is rejected)
How many real solutions does 9x² − 12x + 4 = 0 have? Find them.
Answer: One repeated solution, x = 2/3, because the discriminant is 0
Without solving, find the sum and product of the roots of 2x² − 10x + 3 = 0.
💡 Hint: Sum = −b/a and product = c/a.
Answer: Sum = 5, product = 1.5
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Worked Examples
🌍Quadratic Formula around the world
The same mathematics is taught at different ages, under different names, and assessed in different ways. Here is where this topic sits in the main curricula.
United States
- When it is taught
- Algebra 1, usually grade 8 or 9, then revisited in Algebra 2 with complex roots.
- Where it is assessed
- SAT, ACT, and state assessments such as STAAR, the New York Regents, and CAASPP.
- Worth knowing
- Common Core standard HSA-REI.B.4 asks students to derive the formula by completing the square, not just apply it. Expect questions that ask you to justify the method.
United Kingdom
- When it is taught
- GCSE Higher tier, Years 10 to 11, then extended at A-Level to complex roots in Further Maths.
- Where it is assessed
- GCSE Mathematics with AQA, Edexcel or OCR, then A-Level.
- Worth knowing
- Answers are often required in surd form rather than as decimals. Check your board's formula sheet, since what is provided and what must be recalled varies.
Germany and Austria
- When it is taught
- Klasse 9 to 10, as part of quadratische Gleichungen.
- Where it is assessed
- Mittlerer Schulabschluss and Abitur.
- Worth knowing
- Students normally learn the p-q formula rather than the version with a, b and c. Both give the same answers. See the section on the p-q formula below.
Sweden and the Nordics
- When it is taught
- Årskurs 9, then Matematik 2 in gymnasiet.
- Where it is assessed
- Nationella prov.
- Worth knowing
- The p-q formula, known as pq-formeln, is the standard method. Divide by a first so the x² term has a coefficient of 1.
India
- When it is taught
- Class 10, in the Quadratic Equations chapter.
- Where it is assessed
- CBSE, ICSE and state board examinations.
- Worth knowing
- The formula is often called Shreedharacharya's formula, after the Indian mathematician. Questions frequently ask for the nature of the roots using the discriminant.
Australia and New Zealand
- When it is taught
- Year 10 and Year 10A, then Mathematical Methods in the senior years.
- Where it is assessed
- NAPLAN numeracy at earlier levels, then state certificates and NCEA.
- Worth knowing
- Real-world modelling questions, especially projectile motion and area, appear often.
France
- When it is taught
- Première, in the specialité mathématiques course.
- Where it is assessed
- Baccalauréat.
- Worth knowing
- The discriminant is written as Δ and is central to how the topic is taught. Solutions are called racines, meaning roots.
International Baccalaureate
- When it is taught
- Mathematics: Analysis and Approaches, and Applications and Interpretation, at both SL and HL.
- Where it is assessed
- IB Diploma examinations.
- Worth knowing
- HL extends the topic to complex roots and to the relationship between roots and coefficients.
❓Quadratic Formula FAQ
When should I use the formula instead of factoring?
Try factoring first. If you cannot spot the factors within about 30 seconds, switch to the formula. It always works, even when the answers are decimals or surds.
How do I know how many solutions an equation has?
Check the discriminant, b² − 4ac. A positive value gives two real solutions, zero gives one repeated solution, and a negative value gives no real solutions.
What if a = 0?
Then the equation is not quadratic. With no x² term it is a linear equation, and you solve it by rearranging instead.
Is completing the square the same as using the formula?
They give the same answers. The formula is simply the result of completing the square on ax² + bx + c = 0 once, so you never have to repeat the work.
What does the ± symbol mean?
It means run the calculation twice. Add the square root for one solution, then subtract it for the other.
What is the difference between the quadratic formula and the p-q formula?
They are the same rule in different clothes. The p-q formula, used in Germany and the Nordics, applies to x² + px + q = 0 and reads x = −p/2 ± √((p/2)² − q). Divide your equation by a first and the two methods agree exactly.
Why do word problems often have only one valid answer?
Because the second root is usually impossible in context. A width, a time or a number of items cannot be negative, so that root is discarded even though it satisfies the equation.
Do I have to memorise the formula?
It depends on where you sit your exams. Some boards print it on a formula sheet and others expect recall. Learning it is safer, and singing it to a familiar tune is a method that works for a lot of students.