One expression under the square root sign tells you how many solutions a quadratic has, and what kind, before you solve it.
Today's hook, You can tell whether a quadratic has two solutions, one, or none at all without solving it. The information is sitting inside the square root of the quadratic formula.
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Recall, your gut answer first
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Recall, your gut answer first
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In the quadratic formula, what happens to the answer if the number under the square root is negative? What if it is exactly zero? Predict how many solutions each case gives.
Before you work it out, what is your instinct? Write it down, then check it against the lesson.
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The discriminant is what sits under the root
Work through the core explanation before applying it.
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The discriminant is what sits under the root
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In the quadratic formula the expression $b^2 - 4ac$ sits under the square root sign. Its sign alone decides how many real solutions exist, because a square root behaves differently for positive, zero and negative inputs.
If $\Delta > 0$ the root is a real number and the $\pm$ gives two distinct solutions. If $\Delta = 0$ the $\pm$ adds nothing, so there is exactly one. If $\Delta < 0$ there is no real square root, so there are no real solutions.
$\Delta = b^2 - 4ac$ $\Delta > 0$: two distinct real roots $\Delta = 0$: one real root $\Delta < 0$: no real roots
Signs matter in the substitution
For $b = -5$, $b^2 = 25$, not $-25$. Square the whole coefficient including its sign.
A perfect square means rational roots
If $\Delta$ is a perfect square such as 49, the roots are rational and the quadratic factorises over the integers.
One root still means one value
When $\Delta = 0$ there is a single solution, sometimes described as a repeated or double root.
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What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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What you'll master
Know
Key facts
The discriminant is $\Delta = b^2 - 4ac$.
$\Delta > 0$ gives two distinct real roots, $\Delta = 0$ gives one, $\Delta < 0$ gives none.
If $\Delta$ is a positive perfect square, the roots are rational; otherwise they are irrational.
The discriminant can be used to find unknown coefficients from a condition on the roots.
Understand
Concepts
Why the sign of $\Delta$ controls the number of real roots, via the square root in the formula.
Why $\Delta = 0$ produces exactly one solution rather than two identical answers being counted twice.
Why a perfect square discriminant means the quadratic factorises over the integers.
Can do
Skills
Calculate the discriminant and state the number and nature of the roots.
Distinguish rational from irrational roots using the discriminant.
Find an unknown coefficient given a condition such as equal roots or no real roots.
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Key terms
DiscriminantThe expression $b^2 - 4ac$, which tells you how many real solutions a quadratic has before you solve it. Like this: for $x^2 - 4x + 4 = 0$, $\Delta = 16 - 16 = 0$, so there is exactly one solution.
Distinct rootsTwo different solutions, which happens when the discriminant is positive. Like this: $x^2 - 5x + 6 = 0$ has $\Delta = 1 > 0$ and the two roots 2 and 3.
Equal rootsA single repeated solution, which happens when the discriminant is zero. Like this: $x^2 - 6x + 9 = 0$ has $\Delta = 0$ and the only solution $x = 3$.
No real rootsNo real solution exists, which happens when the discriminant is negative. Like this: $x^2 + x + 5$ has $\Delta = 1 - 20 = -19 < 0$.
Rational rootA root that can be written as a fraction, which happens when the discriminant is a perfect square. Like this: $\Delta = 49$ gives rational roots because $\sqrt{49} = 7$.
Irrational rootA root containing a surd, which happens when the discriminant is positive but not a perfect square. Like this: $\Delta = 12$ gives roots containing $\sqrt{12} = 2\sqrt{3}$.
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What the discriminant tells you
Work through the core explanation before applying it.
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What the discriminant tells you
core concept
Write the quadratic as $ax^2 + bx + c = 0$ and calculate $\Delta = b^2 - 4ac$. You do not need to solve the equation.
The formula contains $\sqrt{\Delta}$. If $\Delta > 0$ that square root is a real number, and the $\pm$ produces two different answers. If $\Delta = 0$ the square root is 0, so $\pm 0$ adds nothing and there is one answer.
If $\Delta < 0$ there is no real square root of a negative number, so the equation has no real solutions at all.
Square the sign too. For $2x^2 - 5x + 3 = 0$, $b = -5$, so $b^2 = 25$. Writing $-25$ here reverses the conclusion and is the most common error in this topic.
Quick check: how many real roots does $x^2 + 2x + 5 = 0$ have?
$\Delta = b^2 - 4ac$. Positive gives two distinct real roots, zero gives exactly one, negative gives none. The sign of $\Delta$ is decided before any solving, and $b$ must be squared with its sign attached.
Pause, copy $\Delta = b^2 - 4ac$, the three cases and what each means, and the warning that $b = -5$ gives $b^2 = +25$, into your book.
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Rational or irrational roots
core concept
We just saw that the sign of the discriminant fixes how many real roots there are. That raises a question: when there are two roots, can you tell whether they will be neat fractions or messy surds? This card answers it → look at whether the discriminant is a perfect square.
If $\Delta$ is a positive perfect square such as 1, 4, 9 or 49, then $\sqrt{\Delta}$ is a whole number, so the roots come out rational and the quadratic factorises over the integers.
If $\Delta$ is positive but not a perfect square, $\sqrt{\Delta}$ is a surd, so the roots are irrational and factorising over the integers will not work.
This is why the discriminant is a good first check: $\Delta = 12$ tells you to stop hunting for factors and go straight to the formula.
Assumes rational coefficients. The perfect-square test tells you about rational roots only when $a$, $b$ and $c$ are themselves rational, which they always are in this course.
Which discriminant value does NOT give rational roots?
A positive perfect square discriminant gives rational roots and means the quadratic factorises over the integers. A positive non-square discriminant gives irrational roots containing a surd, so use the formula rather than hunting for factors.
Pause, copy the perfect-square test for rational roots, and note that $\Delta = 12$ means go straight to the formula, into your book.
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Finding an unknown coefficient
core concept
We just saw how to read the nature of the roots from a known discriminant. That raises a question: what if the roots are described but a coefficient is unknown? This card answers it → turn the description into an equation or inequality in $\Delta$ and solve it.
A condition on the roots becomes a condition on $\Delta$. "Equal roots" means $\Delta = 0$; "no real roots" means $\Delta < 0$; "two distinct real roots" means $\Delta > 0$.
For $x^2 + kx + 9 = 0$ with equal roots: $\Delta = k^2 - 36 = 0$, so $k^2 = 36$ and $k = \pm 6$. Both values are valid answers.
For a condition such as "no real roots" you get an inequality in the unknown, which is often a quadratic inequality. That is the subject of the next lesson.
Do not lose the second value. $k^2 = 36$ gives $k = 6$ and $k = -6$. Writing only $k = 6$ is a half answer.
Fill the blank: for equal roots, the discriminant must equal .
Translate the condition into $\Delta$: equal roots means $\Delta = 0$, no real roots means $\Delta < 0$, two distinct roots means $\Delta > 0$. Then solve for the unknown, keeping both values when a square root is taken.
Pause, copy the three translations of root conditions into discriminant conditions, and the worked $x^2 + kx + 9 = 0$ giving $k = \pm 6$, into your book.
Worked examples · 3 in a row, reveal as you go
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Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · NATURE OF THE ROOTS
Determine the number and nature of the roots of $2x^2 - 5x + 3 = 0$.
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$a = 2$, $b = -5$, $c = 3$
Read coefficients with their signs.
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$\Delta = (-5)^2 - 4(2)(3) = 25 - 24 = 1$
Square $-5$ to get $+25$.
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$\Delta = 1 > 0$ and 1 is a perfect square, so there are two distinct rational roots
Positive means two roots; perfect square means rational.
PROBLEM 2 · FINDING AN UNKNOWN
Find the values of $k$ for which $x^2 + kx + 9 = 0$ has equal roots.
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Equal roots means $\Delta = 0$
Translate the condition into the discriminant.
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$\Delta = k^2 - 4(1)(9) = k^2 - 36$
Substitute $a = 1$, $b = k$, $c = 9$.
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$k^2 - 36 = 0 \Rightarrow k = \pm 6$
Solve, keeping both signs.
PROBLEM 3 · PROVING NO REAL ROOTS
Show that $3x^2 + x + 5 = 0$ has no real roots.
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$a = 3$, $b = 1$, $c = 5$
Read the coefficients.
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$\Delta = 1^2 - 4(3)(5) = 1 - 60 = -59$
Substitute into $b^2 - 4ac$.
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$\Delta = -59 < 0$, so there are no real roots
A negative discriminant has no real square root, so the formula produces no real values.
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Quick-fire practice
Work through the core explanation before applying it.
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Quick-fire practice
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Find $\Delta$ for $x^2 - 6x + 9 = 0$ and state the number of roots.
Find $\Delta$ for $x^2 + 3x - 4 = 0$ and state whether the roots are rational.
Show that $x^2 + x + 1 = 0$ has no real roots.
Find $k$ if $x^2 + 8x + k = 0$ has equal roots.
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Revisit your prediction
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
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Revisit your prediction
At the start you predicted what a negative or zero value under the square root would do. Confirm both predictions using $\Delta$, and explain why $\Delta = 0$ gives one root rather than two identical ones being counted separately.
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Multiple choice
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Multiple choice
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Short answer
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Short answer
ApplyBand 43 marks
Q1. Determine the number and nature of the roots of $3x^2 - 7x + 2 = 0$, showing your discriminant calculation. (3 marks)
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ApplyBand 53 marks
Q2. Find all values of $m$ for which $x^2 + mx + 16 = 0$ has equal roots. (3 marks)
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UnderstandBand 32 marks
Q3. Explain why a quadratic with a negative discriminant has no real roots, referring to the quadratic formula. (2 marks)
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📖 Comprehensive answers (click to reveal)
Practice 1: $\Delta = 36 - 36 = 0$, one root. Practice 2: $\Delta = 9 + 16 = 25$, a perfect square, so two rational roots. Practice 3: $\Delta = 1 - 4 = -3 < 0$. Practice 4: $\Delta = 64 - 4k = 0$, so $k = 16$.
Q1 (3 marks): $a=3$, $b=-7$, $c=2$ [1]. $\Delta = 49 - 24 = 25$ [1]. $\Delta > 0$ and is a perfect square, so two distinct rational roots [1].
Q3 (2 marks): The quadratic formula contains $\sqrt{b^2-4ac}$ [1]. A negative number has no real square root, so the formula produces no real values and the equation has no real solutions [1].
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Review and finish
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Boss battle · Discriminant Detective
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Read the number and nature of the roots from the discriminant, and work backwards to find unknown coefficients. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.