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Solving Linear-Quadratic Systems

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Solving Linear-Quadratic Systems

Study Guide

Key Definition

A linear-quadratic system includes a linear equation $y = mx + d$ (slope-intercept form) and a quadratic equation $y = ax^2 + bx + c$.

Important Notes

Use substitution to solve linear-quadratic systems.
The slope-intercept form of a line is $y = mx + d$ and the standard form is $Ax + By = C$.
The quadratic equation standard form is $y = ax^2 + bx + c$.
Substitute the expression for $y$ from the line into the quadratic.
Solve the resulting quadratic using the quadratic formula or factoring when possible.
You can also solve such systems graphically by plotting both curves and reading their intersection points.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\div$ represents division

$x^2$ indicates x squared

$Δ$ represents the discriminant, $b^2 - 4ac$

$(x,y)$ denotes an intersection point

Remember to use proper notation when solving problems

Why It Works

By substituting the linear equation into the quadratic equation, you create a single-variable quadratic that can be solved using $ax^2 + bx + c = 0$.

Remember

The quadratic formula is crucial in solving $ax^2 + bx + c = 0$ equations.

Quick Reference

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

Understanding Solving Linear-Quadratic Systems

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

To solve a system with a line and a parabola, substitute $y = mx + d$ into $y = ax^2 + bx + c$. Simplify to form $ax^2 + (b - m)x + (c - d) = 0$, solve for $x$, then find $y$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the solution to $y = 2x + 3$ and $y = x^2 + 4x + 5$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard ramp has a linear entry $y = 0.5x + 2$ and a parabolic exit $y = -x^2 + 4x + 3$. Find the intersection points.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the equations $y = -x + 1$ and $y = x^2 + 2x + 1$. Determine the points of intersection.

Challenge Quiz

Single Choice Quiz

Advanced

Given $y = 3x - 4$ and $y = 2x^2 - x + 2$, which is a solution?

Recap

Watch & Learn

Review key concepts and takeaways