The Discriminant and Nature of Roots

Use b² − 4ac to tell whether a quadratic has two different, two equal or no real roots, and find the values of k that give each case.

In the exam for Edexcel Further Pure Maths 4PM1, IGCSE Additional Maths 0606 · 15 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

What b² − 4ac tells you

Before you spend five minutes solving an equation, it helps to know if there is any answer to find. One small number tells you, in ten seconds. It is called the discriminant.

Look at the quadratic formula for $a x^{2} + b x + c = 0$ . Everything depends on the part under the square root.

Formula

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

$a$ — the number in front of $x^{2}$ . It is never 0, or there would be no $x^{2}$ .
$b$ — the number in front of $x$ , with its sign.
$c$ — the number on its own, with its sign.
$b^{2} - 4 a c$ — the discriminant: the part under the square root.

Rule

0606 and 4037 print the quadratic formula in the list of formulas. 4PM1 does not, so learn it. No board prints the discriminant on its own: learn $b^{2} - 4 a c$ .

The square root decides everything. A positive number has a square root, 0 has the square root 0, and a negative number has no real square root.

$b^{2} - 4 a c$	What happens	The roots
Positive	$\pm$ gives two different answers	Two different real roots
Zero	$\pm 0$ gives the same answer twice	Two equal roots
Negative	No real square root	No real roots

Key idea

Around 820 CE in Baghdad, al-Khwarizmi wrote the book that gave us the word algebra. He noticed that some quadratic problems were impossible, when the square of half the x-number was smaller than the plain number. That is the same test you will learn today.

Three cases

The graph of $y = a x^{2} + b x + c$ shows the same three cases. Each real root is an x-value where the curve meets the x-axis.

Formula

b^{2} - 4 a c

$b$ — the number in front of $x$ . Square it with its sign, so $(- 5)^{2} = 25$ .
$a$ — the number in front of $x^{2}$ .
$c$ — the number on its own.

Rule

$b^{2} - 4 a c > 0$ : two different (distinct) real roots. The curve crosses the x-axis twice.
$b^{2} - 4 a c = 0$ : two equal roots, also called a repeated root. The curve just touches the x-axis.
$b^{2} - 4 a c < 0$ : no real roots. The curve never meets the x-axis.
“Real roots” with no other word means $b^{2} - 4 a c \geq 0$ : the first two cases together.

Worked example

1
Show that $3 x^{2} - 4 x + 2 = 0$ has no real roots.
2
$a = 3, b = - 4, c = 2$ — Read the three numbers. Keep the minus sign with b.
3
$b^{2} - 4 a c = (- 4)^{2} - 4 (3) (2)$ — Put them into the discriminant. Brackets keep the signs safe.
4
$= 16 - 24 = - 8$ — Work it out.
5
$- 8 < 0$ — The discriminant is negative, so the equation has no real roots.

Worked example

1
Describe the roots of $4 x^{2} - 12 x + 9 = 0$ .
2
$(- 12)^{2} - 4 (4) (9) = 144 - 144$ — a = 4, b = −12, c = 9.
3
$= 0$ — The discriminant is zero, so there are two equal roots.
4
$x = \frac{12}{8} = \frac{3}{2}$ — With a zero discriminant the formula gives $x = \frac{- b}{2 a}$ only.

Common mistake

Writing $- 4^{2} = - 16$ for b. Square the whole of b, sign and all: $(- 4)^{2} = 16$ . A square is never negative.

Common mistake

Reading a, b and c before the equation equals 0. For $x^{2} + 5 = 3 x$ , first move everything to one side: $x^{2} - 3 x + 5 = 0$ , so $b = - 3$ .

Key idea

Always finish with a sentence: “the discriminant is negative, so there are no real roots”. Mark schemes give the last mark for this conclusion.

Equal roots: find k

Now the quadratic has a letter, k, inside it. The question asks which values of k make the two roots equal. The unknown is now k, not x.

Steps

1
1. Read a, b and c. Some of them contain k, so keep each one in brackets.
2. Write $b^{2} - 4 a c = 0$ .
3. Expand and simplify. You get a quadratic in k.
4. Solve it. You usually get two values of k.

Worked example

1
Find the values of k for which $x^{2} + 2 k x + 3 k - 2 = 0$ has two equal roots.
2
$a = 1, b = 2 k, c = 3 k - 2$ — The whole of $3 k - 2$ is the number on its own.
3
$(2 k)^{2} - 4 (1) (3 k - 2) = 0$ — Equal roots: the discriminant is 0.
4
$4 k^{2} - 12 k + 8 = 0$ — $(2 k)^{2} = 4 k^{2}$ , and $- 4 (3 k - 2) = - 12 k + 8$ .
5
$k^{2} - 3 k + 2 = 0$ — Divide every term by 4.
6
$(k - 1) (k - 2) = 0$ — Factorise.
7
$k = 1 or k = 2$ — Both values work.

Example

Check $k = 2$ : the equation becomes $x^{2} + 4 x + 4 = 0$ , which is $(x + 2)^{2} = 0$ . One repeated root, $x = - 2$ .

Common mistake

Squaring only the letter: $(2 k)^{2}$ is $4 k^{2}$ , not $2 k^{2}$ . In the same way, $(k + 3)^{2}$ is $k^{2} + 6 k + 9$ , not $k^{2} + 9$ .

Common mistake

Losing the brackets on c. $- 4 (3 k - 2)$ is $- 12 k + 8$ . Without brackets you get $- 12 k - 2$ , and every line after it is wrong.

Key idea

Equal roots means discriminant = 0. Write that line first: it earns the method mark even before you expand.

A range of k

Other questions ask for the set of values of k that give two different roots, or no real roots. Now the condition is an inequality, so the answer is a range of k.

Steps

1
1. Write the discriminant with the right sign: $> 0$ , $< 0$ or $\geq 0$ .
2. Expand to a quadratic in k, with all terms on one side.
3. Find the critical values: solve the quadratic in k equal to 0.
4. Sketch the parabola in k. For $> 0$ take the two outside parts. For $< 0$ take the part between.

Worked example

1
Find the set of values of k for which $x^{2} + (k - 3) x + k = 0$ has two distinct real roots.
2
$(k - 3)^{2} - 4 (1) (k) > 0$ — Distinct real roots: discriminant greater than 0.
3
$k^{2} - 6 k + 9 - 4 k > 0$ — Expand the square in full.
4
$k^{2} - 10 k + 9 > 0$ — Collect the k terms.
5
$(k - 1) (k - 9) = 0 \Rightarrow k = 1, 9$ — The critical values.
6
$k < 1 or k > 9$ — The parabola is above 0 outside its roots.

Worked example

1
Find the set of values of k for which $k x^{2} + 4 x + k - 3 = 0$ has no real roots.
2
$4^{2} - 4 k (k - 3) < 0$ — No real roots: discriminant less than 0.
3
$16 - 4 k^{2} + 12 k < 0$ — Expand.
4
$k^{2} - 3 k - 4 > 0$ — Divide by −4. Dividing by a negative number flips the sign.
5
$(k - 4) (k + 1) = 0 \Rightarrow k = - 1, 4$ — The critical values.
6
$k < - 1 or k > 4$ — Greater than 0: the two outside parts.

Rule

Write an “outside” answer as two pieces, like $k < 1$ or $k > 9$ . Write a “between” answer as one, like $- 1 < k < 4$ . Use $\leq$ or $\geq$ only when the question says “real roots”.

Common mistake

Writing $1 > k > 9$ . No number is below 1 and above 9 at the same time. Use two separate pieces.

Common mistake

Dividing by a negative number and keeping the same sign. $- 4 k^{2} + 12 k + 16 < 0$ becomes $k^{2} - 3 k - 4 > 0$ .

Key idea

Each critical value is the k that gives equal roots. The inequality tells you which side of them you need.

True for every k

Some questions ask you to show that something is true for every value of k. You cannot test every k, so you show the discriminant always has one sign.

The key fact: a square is never negative. So $(k - 2)^{2} \geq 0$ , and $(k - 3)^{2} + 4$ is always at least 4.

Worked example

1
Show that $x^{2} + (k + 2) x + 2 k = 0$ has real roots for every value of k.
2
$(k + 2)^{2} - 4 (1) (2 k)$ — Write the discriminant.
3
$= k^{2} + 4 k + 4 - 8 k$ — Expand.
4
$= k^{2} - 4 k + 4 = (k - 2)^{2}$ — This is a perfect square.
5
$(k - 2)^{2} \geq 0$ — A square is never negative, so the roots are always real.

Worked example

1
Show that $x^{2} + (k + 1) x + 2 k - 3 = 0$ has two distinct real roots for every value of k.
2
$(k + 1)^{2} - 4 (2 k - 3) = k^{2} - 6 k + 13$ — Discriminant, expanded and simplified.
3
$= (k - 3)^{2} - 9 + 13$ — Complete the square: half of −6 is −3.
4
$= (k - 3)^{2} + 4$ — Tidy up.
5
$(k - 3)^{2} + 4 > 0$ — A square plus 4 is always at least 4, so there are always two distinct real roots.

Common mistake

Trying a few values of k and saying “it works”. A few examples do not prove it for every k. Complete the square.

Key idea

To show a discriminant is always positive, write it as a square plus a positive number. Then say why that is never 0 or less.

Which test? Read the question

Before you start, read what the question asks. The words tell you which sign to use.

You see	You do
“Describe the nature of the roots”	Work out $b^{2} - 4 a c$ , compare it with 0, write a sentence
“Two equal roots” or “a repeated root”	$b^{2} - 4 a c = 0$
“Two distinct (different) real roots”	$b^{2} - 4 a c > 0$
“Real roots”	$b^{2} - 4 a c \geq 0$
“No real roots” or “is never zero”	$b^{2} - 4 a c < 0$
“For all values of k”	Complete the square on the discriminant

Rule

On 0606 Paper 1 there is no calculator. Critical values may be fractions or surds, so leave them exact, like $k = - \frac{3}{4}$ or $k = 2 \pm 3$ .

Common mistake

Mixing up the letters. The x-roots are what the question is about, but the answer it wants is the values of k.

Key idea

Positive: two roots. Zero: one repeated root. Negative: none. Every question in this lesson is that one sentence.

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