7.6 Quadratic Equations

Quadratic Equations

A quadratic equation is any equation that can be written in the form $ax^2 + bx + c = 0$, where $a \neq 0$. Solving these equations means finding the values of $x$ that make the equation true. In this section, you'll connect the factoring skills from earlier in the unit to actually solving equations.

Zero Product Property

The Zero Product Property states: if the product of two factors equals zero, then at least one of those factors must be zero.

Why does this matter? Once you factor a quadratic equation, this property lets you break one harder equation into two simple ones.

Steps to solve a quadratic equation using the Zero Product Property:

1. Write the equation in standard form (everything on one side, zero on the other)
2. Factor the quadratic expression
3. Set each factor equal to zero
4. Solve each resulting equation

Example: Solve $x^2 + 5x + 6 = 0$

1. Already in standard form
2. Factor: $(x + 2)(x + 3) = 0$
3. Set each factor to zero: $x + 2 = 0$ or $x + 3 = 0$
4. Solve: $x = -2$ or $x = -3$

Both values are solutions (also called roots) of the equation.

Factoring Quadratic Expressions

Factoring means rewriting a polynomial as a product of simpler expressions. For a quadratic $ax^2 + bx + c$:

• Find two numbers whose product is $ac$ and whose sum is $b$
• Use those numbers to rewrite the middle term, then factor by grouping

Example: Factor $2x^2 + 7x + 3$

• You need two numbers with product $2 \times 3 = 6$ and sum $7$. Those numbers are $6$ and $1$.
• Rewrite: $2x^2 + 6x + 1x + 3$
• Group: $(2x^2 + 6x) + (x + 3) = 2x(x + 3) + 1(x + 3)$
• Factor out the common binomial: $(2x + 1)(x + 3)$

Not every quadratic factors neatly with integers. When factoring doesn't work, the quadratic formula provides a solution:

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

The expression $b^2 - 4ac$ is called the discriminant. It tells you how many real solutions exist:

• Positive discriminant: two real solutions
• Zero discriminant: one real solution (a repeated root)
• Negative discriminant: no real solutions

Real-World Quadratic Modeling

Quadratic equations show up in problems involving projectile motion, area, and profit/revenue.

Steps for solving word problems:

1. Identify the unknown variable and the given information
2. Set up a quadratic equation that represents the situation
3. Solve by factoring (or the quadratic formula if needed)
4. Check whether each solution makes sense in context

That last step is easy to overlook but really matters. For example, if you're solving for the time a ball hits the ground and you get $t = -3$ and $t = 5$, only $t = 5$ seconds is meaningful because negative time doesn't apply.

Example: A rectangular garden has a length 3 feet longer than its width. The area is 40 square feet. Find the dimensions.

• Let $w$ = width. Then length = $w + 3$.
• Area equation: $w(w + 3) = 40$, which gives $w^2 + 3w - 40 = 0$
• Factor: $(w + 8)(w - 5) = 0$, so $w = -8$ or $w = 5$
• A width can't be negative, so $w = 5$ feet and the length is 8 feet.

Graphical Representation of Quadratic Equations

The graph of a quadratic equation is a U-shaped curve called a parabola. Key features to know:

• Vertex: the highest or lowest point of the parabola (lowest when $a > 0$, highest when $a < 0$)
• Axis of symmetry: a vertical line through the vertex that divides the parabola into two mirror-image halves
• Roots (x-intercepts): the points where the parabola crosses the x-axis, which correspond to the solutions of the equation

The connection between the graph and the algebra is direct: the x-intercepts of the parabola are the same values you find when you set the equation equal to zero and solve. A parabola that doesn't touch the x-axis means the equation has no real solutions (negative discriminant).

Completing the square is a method for rewriting a quadratic in vertex form $a(x - h)^2 + k$, where $(h, k)$ is the vertex. This form makes it straightforward to identify the vertex and axis of symmetry without graphing.