Everything on one side
Write a quadratic equation as ax² + bx + c = 0.
The prayer hall
A hall must have a carpet area of 300 m². Its length is 1 m more than twice its breadth. Call the breadth x metres, so the length is (2x + 1) metres.
Area = length × breadth = (2x + 1) × x = 2x² + x. So the breadth must satisfy 2x² + x = 300.
Equations like this one come in all shapes. How could we write every one of them in the same way, so that we can compare them?
What this lesson covers
The idea
A quadratic equation is p(x) = 0 with p(x) a polynomial of degree 2; written with terms in descending order of degree, ax² + bx + c = 0 (a, b, c real, a ≠ 0) is its standard form.
The prayer hall
A hall must have a carpet area of 300 m². Its length is 1 m more than twice its breadth. Call the breadth x metres, so the length is (2x + 1) metres.
Area = length × breadth = (2x + 1) × x = 2x² + x. So the breadth must satisfy 2x² + x = 300.
Equations like this one come in all shapes. How could we write every one of them in the same way, so that we can compare them?
- Put every term on one side, with 0 on the other
- Keep the plain number on the right side
- Always keep the x² term alone on the right
Carry it over, tidy up
Tap a term on the right of "=" to carry it to the left: its sign flips. When the right side is 0, press Tidy up to add like terms, put the powers in order, and read a, b and c.
The standard form
A quadratic equation is p(x) = 0 where p(x) has degree 2. Written with the terms in descending order of degree, it is ax² + bx + c = 0, with a, b, c real numbers and a ≠ 0. This is its standard form.
The hall: 2x² + x − 300 = 0, so a = 2, b = 1, c = −300. Each number keeps its sign.
Why a ≠ 0? If a were 0, the x² term would vanish and only bx + c = 0 would be left, which has no x². But b and c can be 0: −x² + 301 = 0 has b = 0 and is still quadratic.
Notes
Standard form: ax² + bx + c = 0, with terms in descending order of degree and a ≠ 0. b and c may be 0.
Check yourself
Which is the standard form of 4x − 3x² + 2 = 0?
Bring 1 − x² + 300 = 0 to standard form −x² + … = 0. What is c, the plain number?
Answer: 301
1 + 300 = 301, so the equation is −x² + 301 = 0. Here a = −1, b = 0 and c = 301.
Write 3x² = 5x + 2 in standard form. What is b, the number in front of x?
Answer: -5
3x² − 5x − 2 = 0. So a = 3, b = −5 and c = −2.
For ax² + bx + c = 0 to be a quadratic equation, which number must not be 0?
- −3x² + 4x + 2 = 0 — correct. Yes! Same terms, in descending order of degree, with 0 on the right.
- 4x − 3x² + 2 = 0. That is the same equation, but the terms are not in descending order. Put the x² term first.
- −3x² + 4x = 2. The standard form has 0 on the right side. Carry the 2 over.
- a — correct. Yes! If a = 0, the x² term vanishes and it is no longer quadratic.
- b. b can be 0: x² − 9 = 0 is quadratic, with b = 0.
- c. c can be 0: x² + 3x = 0 is quadratic, with c = 0.