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Quadratic Formula

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Quadratic Formula

Study Guide

Key Definition

The quadratic formula provides solutions to the quadratic equation $ax^2 + bx + c = 0$.

Important Notes

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Quadratic equations have at most two real solutions.
The discriminant $b^2 - 4ac$ determines the nature of the roots.
If $b^2 - 4ac = 0$, the equation has one real solution.
If $b^2 - 4ac < 0$, the equation has no real solutions but has two complex conjugate solutions.

Mathematical Notation

$\pm$ indicates plus or minus

$\sqrt{x}$ represents the square root of x

$x^2$ indicates x squared

$\frac{a}{b}$ represents a divided by b

$b^2 - 4ac$ is the discriminant

Remember to use proper notation when solving problems

Why It Works

The quadratic formula derives from completing the square of the standard quadratic equation $ax^2 + bx + c = 0$.

Remember

Always check the discriminant $b^2 - 4ac$ to determine the number and type of solutions.

Quick Reference

Quadratic Formula:$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Understanding Quadratic Formula

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ gives the solutions of ax^2 + bx + c = 0 by substituting the coefficients and computing the discriminant.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the solution to the equation x^2 - 4x + 4 = 0 using the quadratic formula?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder jumps off a ramp with an initial velocity of 15 m/s and an initial height of 2 m. Calculate the maximum height reached using h(t) = -4.9t^2 + vt + h_0.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Explore the roots of the equation 2x^2 - 3x + 1 = 0

Challenge Quiz

Single Choice Quiz

Advanced

Find the roots of 3x^2 - 6x + 2 = 0 using the quadratic formula.

Recap

Watch & Learn

Review key concepts and takeaways