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Quadratic equations

How to solve a quadratic equation: factoring, completing the square, the quadratic formula, and the discriminant as the number of real solutions.

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4

lessons

36

practice questions

6

prerequisite lessons

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Quadratics with no real solutions

Learn to identify quadratic equations with no real solutions by computing the discriminant, understand why a negative discriminant means no real square root exists, and connect this to the geometric picture of a parabola that does not cross the xx-axis.

x=−b±Δ2ax = \dfrac{-b \pm \sqrt{\Delta}}{2a}

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Lesson 04 · Quadratics with no real solutions

123456−1−212345678−3−4−1−2
(1.0,4.0)
Δ=−16.0 - no real solutions

y=1.0x2−2.0x+5.0

a 1.0
b −2.0
c 5.0

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The 4 lessons

The 4 lessons in this unit. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.

01

Solving quadratic equations by factoring

12 questions

Learn how to recognise the standard form ax2+bx+c=0ax^2 + bx + c = 0, factor quadratic expressions by finding two numbers with the right sum and product, and solve the equation using the zero product property.

Covers

What is a quadratic equation

Factoring simple quadratics

Solving by factoring

02

The quadratic formula

8 questions

Learn how to solve any quadratic equation using the quadratic formula, and use the discriminant to determine how many real solutions an equation has.

Covers

The quadratic formula

The discriminant

03

Completing the square

8 questions

Learn how to recognise a perfect square trinomial from its coefficients, and rewrite any monic quadratic in completed-square form.

Covers

Perfect square trinomials

Completing the square

04

Quadratics with no real solutions

Preview

8 questions

Learn to identify quadratic equations with no real solutions by computing the discriminant, understand why a negative discriminant means no real square root exists, and connect this to the geometric picture of a parabola that does not cross the xx-axis.

Covers

Negative discriminant means no real solutions

Geometric interpretation

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Where this unit sits in the curriculum

This unit is part of our Mathematical Foundations for ML learning path, which contains 83 lessons. Every one of them is drawn below.

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Every dot is a lesson and every line a prerequisite. The 4 red dots are the lessons in Quadratic equations; the 6 blue dots feeding into them are the lessons Quadratic equations depends on; the 73 pale dots are the rest of the learning path, which come after Quadratic equations or alongside it.Quadratic equationsFirst lessons on the left83 lessons

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4 lessons in this unit

6 prerequisite lessons across the unit, counting every step back to the start

73 other lessons in the learning path - after this unit, or alongside it

Every dot is a lesson, every line a prerequisite. You can only start a lesson once you have mastered all of its prerequisites, so you are always building on solid foundations.

Builds on: Exponents and roots, Inequalities and absolute value

Leads to: Eigenvalues, eigenvectors and singular value decomposition, Limits

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More in Mathematical foundations: Inequalities and absolute value, Set operations, Exponents and roots, Functions, Summation, product, and indexed notation, Logic, Exponentials and logarithms, Trigonometry