🟣 Quadratic Equations

Master advanced techniques for solving quadratic equations! Explore the discriminant, vertex forms, and real-world applications.

📐 Understanding Quadratic Equations (Beyond the Basics)

A quadratic equation is a polynomial equation where the highest power of the variable is 2. It is written in standard form as:

$$ax^2 + bx + c = 0$$

Key Properties

$a, b, c$ are real numbers, and $a \neq 0$.
The highest exponent is 2, so the graph is a parabola.
The parabola opens upward if $a > 0$ and downward if $a < 0$.

Example:

$$3x^2 - 5x + 2 = 0$$

Here: $a=3$, $b=-5$, $c=2$

In Level 1 and Level 2, we covered basic factoring, the quadratic formula, and completing the square. Now, let's explore advanced methods, applications, and graphical interpretations.

🔍 Nature of the Roots – The Role of the Discriminant

The discriminant helps determine the number and type of solutions of a quadratic equation. It is given by:

$$D = b^2 - 4ac$$

Three Scenarios

If $D > 0$: Two real and distinct roots (parabola crosses the x-axis twice).
If $D = 0$: One real repeated root (parabola touches the x-axis at one point).
If $D < 0$: No real solutions (parabola does not touch the x-axis, complex roots).

Example

For $4x^2 - 12x + 9 = 0$:

$$D = (-12)^2 - 4(4)(9) = 144 - 144 = 0$$

Since $D = 0$, there is one repeated root. Solving: $x = \frac{12}{8} = \frac{3}{2}$

📊 Graphing Quadratic Equations (Beyond the Basics)

Every quadratic equation graphs as a parabola. Key features include:

Vertex: The highest or lowest point
Axis of Symmetry: The vertical line through the vertex, given by: $$x = \frac{-b}{2a}$$
X-Intercepts (Roots): Where $y = 0$
Y-Intercept: The value of $y$ when $x = 0$ (always equals $c$)

Example: Graphing

Graph $f(x) = x^2 - 6x + 8$:

Step 1: Find the vertex

$x = \frac{6}{2} = 3$

Then, $f(3) = 9 - 18 + 8 = -1$; Vertex = $(3,-1)$

Step 2: Find intercepts

Y-intercept: $f(0)=8$
X-intercepts: Factor $x^2-6x+8=(x-2)(x-4)$, so $x=2$ and $x=4$

The parabola opens upward since $a=1>0$.

💡 Finding the Vertex by Completing the Square

Another way to find the vertex is to convert the standard form $y = ax^2 + bx + c$ into vertex form $y = a(x-h)^2 + k$, where $(h,k)$ is the vertex.

Example: Completing the Square

Convert $y = x^2 + 6x + 5$ to vertex form.

Group x-terms: $y = (x^2 + 6x) + 5$
Add and subtract $(\frac{6}{2})^2 = 9$ inside the group:

$$y = (x^2 + 6x + 9 - 9) + 5$$

Factor the perfect square and move the constant out:

$$y = (x+3)^2 - 9 + 5 = (x+3)^2 - 4$$

The vertex is $(-3, -4)$.

🔧 Advanced Factoring Techniques for Quadratics

Factoring When $a \neq 1$ (Decomposition Method)

When the leading coefficient isn't 1, use the decomposition method:

Example

Factor $6x^2 - 7x - 3$

Step 1: Multiply $a$ and $c$: $6 \times (-3) = -18$
Step 2: Find two numbers that multiply to -18 and add to -7: (-9 and 2)
Step 3: Rewrite: $6x^2 - 9x + 2x - 3$
Step 4: Group: $(6x^2 - 9x) + (2x - 3)$
Step 5: Factor: $3x(2x-3) + 1(2x-3) = (3x+1)(2x-3)$

Final Answer: $(3x+1)(2x-3)$

Vieta's Formulas

For a quadratic equation $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$:

Sum of roots: $r_1 + r_2 = -\frac{b}{a}$
Product of roots: $r_1 \times r_2 = \frac{c}{a}$

Example: For $2x^2 - 6x + 4 = 0$

Sum = $-\frac{-6}{2} = 3$ and Product = $\frac{4}{2} = 2$

🚀 Solving Quadratics in Real-World Applications

Quadratic equations are used in fields such as physics, business, and engineering.

Example: Projectile Motion

A ball is thrown upward with an initial velocity of 20 m/s. Its height at time $t$ seconds is given by:

$$h(t) = -5t^2 + 20t + 30$$

Set $h(t)=0$ to find when the ball hits the ground:

$$t = \frac{-20 \pm \sqrt{400+600}}{-10} = \frac{-20 \pm \sqrt{1000}}{-10}$$

Final Answer: $t \approx 5.16$ seconds

Example: Business Revenue

A company's revenue function is $R(x) = -2x^2 + 80x - 200$ where $x$ is units sold in thousands.

Find the production level that maximizes revenue:

Vertex: $x = \frac{-80}{2(-2)} = 20$

Maximum revenue occurs at 20,000 units.

🎯 Practice Questions

Master these fundamental concepts!

Find the number of solutions for: $5x^2 - 6x + 2 = 0$

Graph $y = -x^2 + 4x - 3$ and find its vertex

Solve using factoring: $4x^2 - 9x + 2 = 0$

Solve using the quadratic formula: $x^2 + 5x - 7 = 0$

Find the range of: $f(x) = -2x^2 + 4x + 5$

Solve for $x$ using completing the square: $x^2 + 10x + 21 = 0$

A rectangular garden has an area of 120 m² and its length is 4 m more than its width. Find its dimensions.

The revenue function is $R(x)=-2x^2+20x$. Find the number of items that maximize revenue.

Find the value of $k$ for which $x^2+kx+9$ has only one solution.

For $h(t)=-4t^2+16t+50$, find when the rocket reaches maximum height.

🔥 Challenge Questions

Test your advanced skills!

Prove that the sum of the roots of $ax^2+bx+c=0$ is $-\frac{b}{a}$ and the product is $\frac{c}{a}$.

Solve for $x$: $\frac{x^2+5x+6}{x+2}=0$

Find the equation of a parabola with vertex $(2,-3)$ passing through $(0,5)$.

A football is kicked and its height is given by $h(t)=-5t^2+25t$. Find when the ball reaches maximum height.

Find the integer solutions for $x^2-7x+12=0$.