Quadratic equations: Formulas

Class 10

The quadratic formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

What each letter means
$x$	a root: a value of x that makes ax² + bx + c = 0 true
$a$	the coefficient of x² (not 0)
$b$	the coefficient of x
$c$	the constant term

Every quadratic equation ax² + bx + c = 0 can be solved with one formula: the ± gives the two roots, one with + and one with −.

Why it works

Divide by a and complete the square: x² + (b/a)x = −c/a becomes (x + b/2a)² = (b² − 4ac) ÷ 4a². Taking the square root of both sides gives x + b/2a = ±√(b² − 4ac) ÷ 2a, and moving b/2a across gives the formula. The Indian mathematician Sridharacharya derived it around 1025 CE.

When to use it

Any quadratic equation, especially one that does not factorise easily or has roots that are not whole numbers.

How to use it

Write the equation as ax² + bx + c = 0 and read off a, b and c with their signs.
Work out b² − 4ac (the discriminant); if it is negative there are no real roots.
Take its square root, then work out (−b + √…) ÷ 2a and (−b − √…) ÷ 2a.
Check a root by putting it back into the equation.

To remember: Minus b, plus or minus the root of b squared minus 4ac, all over 2a.

Other forms

Same formula, another formThe two roots add to −b/a and multiply to c/a, the relations between zeroes and coefficients.
Special case $x = - \frac{b}{2 a}$ When b² − 4ac = 0: the two roots are equal (one repeated root).
Special caseWhen b² − 4ac is negative there is no real square root, so the equation has no real roots.When b² − 4ac < 0.

Worked examples

Story

A prayer hall's floor is a rectangle with area 300 m², and its length is 1 m more than twice its breadth. Find its breadth and length.

If the breadth is x m, the length is 2x + 1 m, so x(2x + 1) = 300, which is 2x² + x − 300 = 0.
$b^{2} - 4 a c = 1 + 2400 = 2401$
$2401 = 49$
$x = \frac{- 1 + 49}{4} = 12$
The other root, (−1 − 49) ÷ 4 = −12.5, cannot be a length. So the breadth is 12 m and the length is 25 m.

Answer: breadth 12 m, length 25 m

Picture

A pole stands on the edge of a circular park of diameter 13 m, at the point where its distances from two gates at opposite ends of a diameter differ by 7 m. How far is it from each gate?

The diameter makes a right angle at the pole, so x² + (x + 7)² = 13², which simplifies to x² + 7x − 60 = 0.
$b^{2} - 4 a c = 49 + 240 = 289$
$x = \frac{- 7 + 17}{2} = 5$
The other root, −12, cannot be a distance. So the pole is 5 m from one gate and 12 m from the other.

Answer: 5 m and 12 m

Direct

Solve 2x² − 5x + 3 = 0 with the quadratic formula.

$b^{2} - 4 a c = 25 - 24 = 1$
$x = \frac{5 + 1}{4} = \frac{3}{2}$
$x = \frac{5 - 1}{4} = 1$

Answer: x = 1 or x = 3/2

Reverse

One root of x² − 6x + c = 0 is 3 + √2. Find c and the other root.

Try it first, then show the working

The formula gives x = 3 ± √(9 − c), so √(9 − c) = √2, and c = 7.
The other root takes the minus sign: 3 − √2.
Check: the roots add to 6 = −(−6) ÷ 1 and multiply to 9 − 2 = 7 = c.

Answer: c = 7; the other root is 3 − √2

Exam

Find the roots of 3x² − 2√6 x + 2 = 0.

Try it first, then show the working

Here a = 3, b = −2√6 and c = 2, so b² − 4ac = 24 − 24 = 0.
The roots are equal: x = 2√6 ÷ 6 = √6 ÷ 3.

Answer: x = √6/3 (twice)

Common mistake: Dividing only the square root by 2a instead of the whole top, or losing the sign of b (−b is positive when b is negative).

Class 10

The discriminant (how many real roots)

D = b^{2} - 4 a c

What each letter means
$D$	the discriminant of ax² + bx + c = 0
$a$	the coefficient of x² (not 0)
$b$	the coefficient of x
$c$	the constant term

The number b² − 4ac, the part under the square root in the quadratic formula, tells you how many real roots there are before you solve: two if it is positive, one repeated root if it is zero, none if it is negative.

Why it works

The roots are (−b ± √D) ÷ 2a. A positive D has two square roots, giving two different roots; D = 0 makes + and − give the same root; a negative D has no real square root, so there is no real root.

When to use it

Deciding whether a problem can be solved at all (can this park, this pole, this field exist?), or finding the value of a letter that makes the roots equal.

How to use it

Write the equation as ax² + bx + c = 0.
Work out b² − 4ac.
Positive: two real roots. Zero: two equal roots. Negative: no real roots.

To remember: Positive two, zero one, negative none.

Other forms

Same formula, another formIn the quadratic formula the discriminant sits under the square root: x = (−b ± √D) ÷ 2a.
Special caseA negative discriminant means no real roots: x² + x + 1 = 0 has D = −3 and no real solution.When D < 0.

Worked examples

Story

A pole must stand on the edge of a circular park of diameter 13 m so that its distances from two opposite gates differ by 7 m, which leads to x² + 7x − 60 = 0. Is it possible?

$D = 7^{2} - 4 \times 1 \times (- 60)$
$= 49 + 240 = 289$
D is positive, so there are two real roots and the pole can be placed.

Answer: 289

Picture

Can a rectangular park have a perimeter of 80 m and an area of 400 m²? If its length is x m, then x(40 − x) = 400, which is x² − 40x + 400 = 0.

$D = (- 40)^{2} - 4 \times 1 \times 400$
$= 1600 - 1600 = 0$
D = 0: exactly one answer. The park is possible, and it is a square of side 20 m.

Answer: 0

Direct

Find the discriminant of 2x² − 4x + 3 = 0 and say what it tells you.

$D = (- 4)^{2} - 4 \times 2 \times 3$
$= 16 - 24 = - 8$
The discriminant is negative, so the equation has no real roots.

Answer: −8

Reverse

For which values of k does 2x² + kx + 3 = 0 have two equal roots?

Try it first, then show the working

Equal roots need D = 0: k² − 4 × 2 × 3 = 0.
$k^{2} = 24$
k = 2√6 or k = −2√6.

Answer: k = 2√6 or k = −2√6

Exam

Find the discriminant of 3x² − 2x + 1/3 = 0, and hence find the nature of its roots. Find them if they are real.

Try it first, then show the working

D = (−2)² − 4 × 3 × 1/3 = 4 − 4 = 0, so the two roots are equal.
x = −b ÷ 2a = 2 ÷ 6 = 1/3.

Answer: D = 0: two equal roots, x = 1/3

Common mistake: Squaring b without its sign mattering (b² is never negative, even when b is), or forgetting that −4ac becomes +4|ac| when a and c have opposite signs.