Standard form
Quadratic equation — An equation of one variable in which the highest power of that variable is 2. Its standard form is ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0.
The condition a ≠ 0 is part of the definition. If a were zero the x² term would vanish and the equation would be linear, not quadratic.
Before applying any method, the equation must be arranged into standard form: all terms on one side, zero on the other, written in descending powers of x.
- a
- coefficient of x², called the leading coefficienta ≠ 0
- b
- coefficient of x
- c
- constant termthe value of y where the curve meets the y-axis
Write 3x(x − 2) = 5 − x in standard form and state a, b and c.
- Expand the left side:
3x² − 6x = 5 − x.Remove brackets before collecting terms. - Move every term to the left:
3x² − 6x − 5 + x = 0.Standard form requires zero on the right. - Collect like terms:
3x² − 5x − 5 = 0.−6x + x = −5x.
a = 3, b = −5, c = −5
The graph of a quadratic
Parabola — The curve produced by the graph of y = ax² + bx + c. It is symmetrical about a vertical line through its turning point, called the axis of symmetry.
Every quadratic graph has exactly one turning point. The sign of a decides its type:
The roots of ax² + bx + c = 0 are the values of x where the parabola meets the x-axis. A quadratic therefore has at most two real roots.
| Condition | Parabola opens | Turning point is a |
|---|---|---|
| a > 0 | upward | minimum |
| a < 0 | downward | maximum |
Move a through zero: the parabola flips and flattens. Move c: the curve slides vertically, since c is the y-intercept. Move b: the vertex shifts sideways as well as vertically.
Methods of solution
Three standard methods are available. All three give the same roots; they differ in speed and in when they can be applied.
| Method | Use it when | Limitation |
|---|---|---|
| Factorisation | the factors are readily seen | only works for factorisable expressions |
| Completing the square | the vertex or the form (x + p)² = q is needed | slower; fractions appear when a ≠ 1 |
| Quadratic formula | any quadratic, including those with irrational roots | none — but slower than factorising |
Solve x² − 7x + 12 = 0.
- Find two numbers whose product is
c = 12and whose sum isb = −7.For a = 1 the factors of the constant that add to b give the split directly. - The numbers are
−3and−4.(−3)(−4) = 12 and (−3) + (−4) = −7. - Factorise:
(x − 3)(x − 4) = 0. - Apply the zero-product rule:
x − 3 = 0orx − 4 = 0.If a product is zero then at least one factor is zero.
x = 3 or x = 4
The quadratic formula
The formula is completing the square carried out once in general, so that the working never has to be repeated. It applies to every quadratic in standard form, whether or not the expression factorises.
- ±
- gives the two roots: one with +, one with −
- b² − 4ac
- the discriminant, written Δ or Dsee the next section
Solve 2x² + 3x − 2 = 0 using the formula.
- Identify the coefficients:
a = 2,b = 3,c = −2.The equation is already in standard form. - Compute the discriminant:
Δ = 3² − 4(2)(−2) = 9 + 16 = 25.Take care with the signs: −4 × 2 × (−2) is +16, not −16. - Substitute:
x = (−3 ± √25) / (2 × 2) = (−3 ± 5) / 4. - Take each sign in turn:
x = 2/4orx = −8/4.
x = ½ or x = −2
The discriminant
Discriminant — The quantity Δ = b² − 4ac, the expression under the square root in the quadratic formula. Its sign determines the nature of the roots without the roots being calculated.
A question that asks how many roots an equation has, or asks for a value of k giving equal roots, or asks whether a line meets a curve, is a discriminant question. In each case the roots themselves are not required.
| Value of Δ | Nature of roots | The graph |
|---|---|---|
| Δ > 0 | two distinct real roots | cuts the x-axis at two points |
| Δ = 0 | one repeated real root | touches the x-axis at one point |
| Δ < 0 | no real roots (two complex roots) | does not meet the x-axis |
| Δ is a perfect square | two distinct rational roots | the expression factorises over the integers |
Find the value of k for which x² + kx + 9 = 0 has equal roots.
- Equal roots require
Δ = 0.This is the condition in row two of the table above. - Here
a = 1,b = k,c = 9, soΔ = k² − 4(1)(9) = k² − 36. - Set
k² − 36 = 0, givingk² = 36. - Take the square root of both sides, keeping both signs.A squared unknown always yields two values; omitting the negative one loses a valid answer.
k = 6 or k = −6
Sum and product of the roots
If α and β are the roots of ax² + bx + c = 0, their sum and product can be written directly from the coefficients. Neither root needs to be found first.
These relations are used to check an answer, and to construct a quadratic when its roots are known.
- α, β
- the two roots of the equation
- −b/a
- sum of the rootsnote the negative sign
- c/a
- product of the roots
Form the quadratic equation whose roots are 2 and −5.
- Sum of roots:
2 + (−5) = −3. - Product of roots:
2 × (−5) = −10. - Substitute into
x² − (sum)x + (product) = 0:x² − (−3)x + (−10) = 0. - Simplify the signs.−(−3) = +3.
x² + 3x − 10 = 0
Common mistake: dividing both sides by x
Given x² = 5x, dividing by x gives x = 5 and loses the root x = 0. Division by a variable is only valid when that variable is known to be non-zero. Instead, bring all terms to one side and factorise: x² − 5x = 0 gives x(x − 5) = 0, so x = 0 or x = 5.
Key points
- Standard form is
ax² + bx + c = 0witha ≠ 0. Rearrange before doing anything else. - The sign of
adecides whether the parabola opens upward (minimum) or downward (maximum). - Three methods of solution: factorisation, completing the square, and the quadratic formula. The formula always works.
- The discriminant
Δ = b² − 4acgives the nature of the roots without solving: positive → two real roots, zero → one repeated root, negative → no real roots. - Sum of roots
= −b/a; product of roots= c/a. Use these to check answers and to construct equations. - Never divide an equation by the variable — factorise instead, or a root is lost.