Solve quadratic equations and inequalities and interpret the discriminant

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Functions

IB Mathematics: Analysis and Approaches SL — Study topic SL 2.7

A quadratic expression has the form

ax^2+bx+c

, where

a

,

b

, and

c

are constants and

a\ne 0

. Solving an equation means finding the values of the variable that make the two sides equal. Solving an inequality means finding all values that make the comparison true. These solutions can be understood algebraically and as points or intervals on a graph. Before starting, recall that a product is zero when at least one factor is zero, and that multiplying or dividing an inequality by a negative number reverses its direction.

What you will learn

Recognise a quadratic equation or inequality and identify its coefficients.
Solve quadratic equations by factorising or using the quadratic formula.
Solve quadratic inequalities by finding the roots and checking the sign of the quadratic between and beyond them.
Use the discriminant to determine how many distinct real roots a quadratic has and connect this to its graph.

1. Equations: factorising and the quadratic formula

First write the equation in the form

ax^2+bx+c=0

. The numbers

a

,

b

, and

c

are the coefficients of the squared term, the linear term, and the constant term. Keeping track of their signs is essential.

When the expression factorises, use the zero-product rule. For example, if

(x-r)(x-s)=0

, then

x=r

or

x=s

. If factorising is not straightforward, use the quadratic formula, which works for every quadratic equation with

a\ne0

. The discriminant,

\Delta=b^2-4ac

, appears under the square root.

The discriminant describes the real solutions. If

\Delta>0

, there are two distinct real roots; if

\Delta=0

, there is one repeated real root; and if

\Delta<0

, there are no real roots. In this lesson, solutions refer to real values of

x

.

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},\qquad \Delta=b^2-4ac

Arrange the equation so one side is zero before identifying coefficients.
Check a proposed root by substituting it into the original equation.
If a calculator gives decimal roots, retain sufficient precision during working and round only as requested.

2. Inequalities: use roots and signs

A quadratic inequality asks where a graph is above or below the horizontal axis. Start by finding the roots of the corresponding equation

ax^2+bx+c=0

. The roots divide the number line into intervals. Within each interval, the expression keeps the same sign, so testing one value in each interval determines the sign throughout that interval.

The leading coefficient tells the end behaviour of the parabola: when

a>0

, it opens upward; when

a<0

, it opens downward. With two distinct roots, an upward-opening quadratic is positive outside the roots and negative between them. For a downward-opening quadratic, these signs are reversed.

Include a root when the inequality allows equality, as in

\geq

or

\leq

. Exclude it for strict inequalities,

>

or

<

. Write the answer as intervals or inequalities, and make clear whether endpoints are included.

ax^2+bx+c\gtrless 0

Solve the matching equation first; the roots are boundary points for the solution intervals.
Test a value in each interval, or use the opening direction and root pattern.
A graphing calculator can display intercepts and the sign of the graph, but use the equation and interval reasoning to justify the answer.

3. Discriminant and multiple representations

On a graph, real roots are the points where the parabola crosses or touches the horizontal axis. Two distinct roots correspond to two crossings. A repeated root corresponds to the graph touching the axis at its turning point. If there are no real roots, the graph does not meet the axis.

The discriminant gives this information without first calculating the roots. For instance, a positive discriminant means the graph crosses the axis twice. A zero discriminant means it just touches, and a negative discriminant means it has no horizontal-axis intercepts.

For a numerical check, substitute each calculated root into the original equation; the result should be zero, apart from possible rounding. For a graphical check, enter the quadratic into a graphing tool and inspect its intercepts. Use a suitable window: a poor viewing window can hide an intercept or make two nearby intercepts difficult to distinguish. The algebraic result remains the justification.

In a context,

x

may represent a quantity such as time or length, so apply any stated restrictions. A purely mathematical solution might include negative values, but a context can rule them out. State the units when the variable represents a measured quantity.

\Delta>0:2\text{ roots},\quad \Delta=0:1\text{ repeated root},\quad \Delta<0:0\text{ real roots}

The discriminant counts distinct real roots: two, one, or none.
Graphing technology is useful for checking roots and intervals, not for replacing algebraic reasoning.
Interpret solutions in the domain specified by the problem.

Worked example

Solve by factorising

Solve

x^2-5x+6=0

.

Factor the quadratic
Find two numbers whose product is $6$ and whose sum is $-5$ . They are $-2$ and $-3$ , so the expression factors as shown.
$x^2-5x+6=(x-2)(x-3)$
Use the zero-product rule
If two factors multiply to zero, at least one factor must be zero. Set each factor equal to zero to obtain both solutions. (x-2)(x-3)=0 \Longrightarrow x=2 or x=3
Check the roots
Substitution confirms that each value makes the original expression zero.
$2^2-5(2)+6=0,\qquad 3^2-5(3)+6=0$

Answer:

x=2

or

x=3

.

Check: The graph of

y=x^2-5x+6

crosses the horizontal axis at

x=2

and

x=3

, matching the algebraic solutions.

Worked example

Solve a quadratic inequality

Solve

x^2-x-6\leq 0

for real

x

.

Find the boundary roots
Factor the corresponding equation. The roots split the number line into three intervals.
$x^2-x-6=(x-3)(x+2),\qquad x=-2,\;3$
Determine where the expression is negative
Test a value between the roots, such as $x=0$ . The expression is $-6$ , so it is negative there. Since the parabola opens upward, it is also positive outside the two roots.
$0^2-0-6=-6<0$
Apply the equality condition
The inequality includes equality, so both roots are part of the solution. The expression is at or below zero between them.
$-2\leq x\leq 3$

Answer: x∈[-2,3].

Check: The graph is on or below the horizontal axis from its intercept at

-2

to its intercept at

3

, inclusive.

Worked example

Interpret the discriminant

Determine how many distinct real roots

2x^2+4x+5=0

has. Explain what this means for its graph.

Identify the coefficients
Compare the equation with $ax^2+bx+c=0$ to read off the three coefficients.
$a=2,\qquad b=4,\qquad c=5$
Calculate the discriminant
Use $\Delta=b^2-4ac$ . A negative result means there are no real roots.
$\Delta=4^2-4(2)(5)=16-40=-24$
Connect to the graph
Because the discriminant is negative, there are no real horizontal-axis intercepts. Since the coefficient of $x^2$ is positive, the parabola opens upward and remains above the axis.
$\Delta<0,\qquad a>0$

Answer: There are no distinct real roots; the graph does not meet the horizontal axis.

Check: A graphing tool should show an upward-opening parabola with no horizontal-axis intercepts.

Common mistakes and how to avoid them

Using the quadratic formula with a coefficient copied from the equation before rearranging it.

Correction: Write the equation as

ax^2+bx+c=0

first, then identify the signed values of

a

,

b

, and

c

.

Giving only the roots when solving an inequality.

Correction: Roots mark interval boundaries; determine which intervals satisfy the inequality and include endpoints only when equality is allowed.

Treating a zero discriminant as two distinct roots.

Correction: A zero discriminant gives one repeated real root, so the parabola touches the axis at one point.

Reversing an inequality when multiplying by a positive number, or forgetting to reverse it when multiplying by a negative number.

Correction: The direction reverses only when multiplying or dividing both sides by a negative number.

Lesson summary

For equations, rearrange to zero and solve by factorising or using the quadratic formula.
Use $\Delta=b^2-4ac$ to determine the number of distinct real roots.
For inequalities, find roots, determine signs on the resulting intervals, and handle endpoints according to the inequality symbol.
Use a graph or calculator as a check, while explaining the algebraic reasoning and any contextual domain restrictions.

Check your understanding

Question 1

How many distinct real roots does

x^2+2x+1=0

have?

No real roots
One repeated real root
Two distinct real roots
correctIndex': 1, ''explanation'': ''The discriminant is $2^2-4(1)(1)=0$ , so there is one repeated real root.''

Show answer and explanation

One repeated real root

The discriminant is

2^2-4(1)(1)=0

, so there is one repeated real root.

Question 2

Solve

x^2-4>0

over the real numbers.

$-2<x<2$
$x<-2$ or $x>2$
$x\leq-2$ or $x\geq2$
correctIndex': 1, ''explanation'': ''The roots are $-2$ and $2$ . The upward-opening quadratic is positive outside the roots, and strict inequality excludes the roots.''

Show answer and explanation

x<-2

or

x>2

The roots are

-2

and

2

. The upward-opening quadratic is positive outside the roots, and strict inequality excludes the roots.

Question 3

What does a negative discriminant tell you about a quadratic graph?

It crosses the horizontal axis twice.
It touches the horizontal axis once.
It has no horizontal-axis intercepts.
correctIndex': 2, ''explanation'': ''A negative discriminant means the quadratic equation has no real roots, so the graph has no horizontal-axis intercepts.''

Show answer and explanation

It has no horizontal-axis intercepts.

A negative discriminant means the quadratic equation has no real roots, so the graph has no horizontal-axis intercepts.

Key terms

Quadratic: An expression or equation involving a squared variable, with a non-zero coefficient of the squared term.
Root: A value of the variable that makes a quadratic expression equal to zero.
Discriminant: The quantity $b^2-4ac$ in the quadratic formula, which determines the number of distinct real roots.
Repeated root: A root that occurs twice as a factor; the graph touches the horizontal axis at that root.

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Published by DoAssignment. This reviewed lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 2.7. It is a study resource, not an official curriculum publication.

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