X-Intercepts Of A Quadratic Function - Varsity Tutors

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x-intercepts of a Quadratic Function

Study Guide

Key Definition

The x-intercepts of a quadratic function $f(x) = ax^2 + bx + c$ are the solutions to the equation $ax^2 + bx + c = 0$.

Important Notes

  • The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ gives the solutions for x-intercepts.
  • If $b^2 - 4ac > 0$, the function has two real and distinct x-intercepts.
  • If $b^2 - 4ac = 0$, the function has one real x-intercept.
  • If $b^2 - 4ac < 0$, the function has no real x-intercepts.
  • The expression $b^2 - 4ac$ is called the discriminant.

Mathematical Notation

$-$ represents subtraction$\pm$ represents plus or minus$\sqrt{a^2 + b^2}$ represents the square root of a^2 + b^2$\frac{p}{q}$ represents the fraction p over qRemember to use proper notation when solving problems

Why It Works

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

Remember

The discriminant $b^2 - 4ac$ helps determine the number and type of roots.

Quick Reference

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

Understanding x-intercepts of a Quadratic Function

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concept. Use space or enter to play video.concept thumbnailBeginner

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Beginner Explanation

To find the x-intercepts of $f(x)=ax^2+bx+c$, set the function equal to zero and factor the quadratic if possible. For example, $x^2-5x+6=(x-2)(x-3)=0$, so x = 2 or x = 3.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

What are the x-intercepts of the quadratic function $f(x) = x^2 - x - 6$?

A$x = -2, x = 3$B$x = 1, x = -6$C$x = 2, x = -3$D$x = -1, x = 6$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

A ball is thrown upwards, and its height is modeled by $h(t) = -16t^2 + 32t + 48$. When does it hit the ground?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

What happens to the x-intercepts if the discriminant is less than zero in $ax^2 + bx + c = 0$?

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Find the x-intercepts of $f(x) = 2x^2 - 4x + 1$.

A$x = 1 - \frac{\sqrt{2}}{2}, x = 1 + \frac{\sqrt{2}}{2}$B$x = \frac{-1}{2}, x = \frac{3}{2}$C$x = \frac{1}{2}, x = 2$D$x = -1, x = 1$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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