Table of Contents
Understanding Quadratic Equations
In this chapter, we focus on equations that involve a squared term and how to recognize and work with them in standard algebraic form. Methods for solving these equations (factoring, completing the square, quadratic formula) are each treated in their own later sections, so here we concentrate on what makes an equation “quadratic” and what its basic features are.
What Makes an Equation Quadratic?
A quadratic equation in one variable is any equation that can be rewritten in the form
$$
ax^2 + bx + c = 0
$$
where:
- $x$ is the variable,
- $a$, $b$, and $c$ are real numbers (called coefficients),
- $a \neq 0$ (this ensures that the $x^2$ term is really present).
The expression $ax^2 + bx + c$ is a quadratic polynomial in $x$. When we set it equal to $0$, we get a quadratic equation.
Examples of quadratic equations:
- $x^2 - 5x + 6 = 0$ (here $a=1$, $b=-5$, $c=6$)
- $3x^2 + 2x - 7 = 0$ (here $a=3$, $b=2$, $c=-7$)
- $-4x^2 + 9 = 0$ (here $a=-4$, $b=0$, $c=9$)
An equation may not look like this at first, but can be rewritten in this form.
Example: $2x(x-3) = 5$ is not in standard form. Expanding and moving everything to one side:
$$
2x^2 - 6x = 5 \\
2x^2 - 6x - 5 = 0
$$
Now it is a quadratic equation with $a=2$, $b=-6$, $c=-5$.
If an equation can be rearranged so that the highest power of $x$ is $2$ and there is no higher power (like $x^3$), then it is quadratic.
Non-examples:
- $5x - 3 = 0$ is linear, not quadratic (highest power is $1$).
- $x^3 - 4x = 0$ is cubic, not quadratic (highest power is $3$).
- $\dfrac{1}{x} + x = 0$ is not a polynomial equation in quadratic form (it has $x$ in the denominator).
Standard Form and Identifying $a$, $b$, and $c$
The standard form of a quadratic equation is:
$$
ax^2 + bx + c = 0
$$
To write an equation in standard form:
- Expand any parentheses.
- Move all terms to one side so the other side is $0$.
- Combine like terms.
- Make sure the $x^2$ term is written first, then the $x$ term, then the constant.
Examples:
- $x(x+4) = 12$
$$
x^2 + 4x = 12 \\
x^2 + 4x - 12 = 0
$$
Here, $a=1$, $b=4$, $c=-12$.
- $5 - 2x^2 = 3x$
$$
5 - 2x^2 - 3x = 0 \\
-2x^2 - 3x + 5 = 0
$$
Here, $a=-2$, $b=-3$, $c=5$.
Sometimes it is convenient to multiply the whole equation by $-1$ to get a positive $a$:
$$
2x^2 + 3x - 5 = 0
$$
Now $a=2$, $b=3$, $c=-5$.
- $7 = x^2$
$$
x^2 - 7 = 0
$$
Here, $a=1$, $b=0$, $c=-7$.
Recognizing $b=0$ or $c=0$ will be useful later when choosing a solving strategy.
Types of Quadratic Equations (by Missing Terms)
Even though all quadratic equations involve $x^2$, they can have different patterns depending on which other terms are present.
- General quadratic (all three terms present):
$$
ax^2 + bx + c = 0, \quad a \neq 0,\ b \neq 0,\ c \neq 0
$$
Example: $2x^2 - 5x + 3 = 0$.
- Incomplete quadratic (no linear term):
$$
ax^2 + c = 0 \quad\text{(here } b=0\text{)}
$$
Example: $3x^2 - 12 = 0$.
- Incomplete quadratic (no constant term):
$$
ax^2 + bx = 0 \quad\text{(here } c=0\text{)}
$$
Example: $x^2 - 4x = 0$.
These special forms often allow simpler solving methods than the most general case. Recognizing the type helps you choose a strategy efficiently later on.
Solutions (Roots) of a Quadratic Equation
A solution (or root) of a quadratic equation $ax^2 + bx + c = 0$ is any value of $x$ that makes the equation true when substituted.
Quadratic equations typically have:
- Two solutions,
- Or one repeated solution,
- Or no real solutions (in which case there are two complex solutions, studied later in complex numbers).
You will learn different methods to find these solutions in other sections; here, we focus on the idea that:
- Solving $ax^2 + bx + c = 0$ means finding all $x$ that satisfy the equation.
Example: $x^2 - 5x + 6 = 0$.
If we test $x=2$:
$$
2^2 - 5\cdot2 + 6 = 4 - 10 + 6 = 0
$$
So $x=2$ is a root. If we test $x=3$:
$$
3^2 - 5\cdot3 + 6 = 9 - 15 + 6 = 0
$$
So $x=3$ is also a root. The solutions are $x=2$ and $x=3$.
You will later see that the graph of $y = x^2 - 5x + 6$ crosses the $x$-axis exactly at these $x$-values.
The Discriminant and Number of Real Solutions
The discriminant of a quadratic equation
$$
ax^2 + bx + c = 0
$$
is the value
$$
\Delta = b^2 - 4ac
$$
(you may also see it written as $D$ instead of $\Delta$).
The sign of the discriminant (positive, zero, or negative) tells you how many real solutions the quadratic equation has:
- If $\Delta > 0$: two distinct real solutions.
- If $\Delta = 0$: one real solution (a repeated root).
- If $\Delta < 0$: no real solutions (two complex solutions).
You will see exactly how this comes from the quadratic formula, but you can already use this to predict the number of real roots without actually solving.
Examples:
- $x^2 - 4x + 3 = 0$
Here $a=1$, $b=-4$, $c=3$.
$$
\Delta = (-4)^2 - 4\cdot1\cdot3 = 16 - 12 = 4 > 0
$$
So there are two distinct real solutions.
- $x^2 - 2x + 1 = 0$
Here $a=1$, $b=-2$, $c=1$.
$$
\Delta = (-2)^2 - 4\cdot1\cdot1 = 4 - 4 = 0
$$
So there is exactly one real solution (a double root).
- $x^2 + 4x + 13 = 0$
Here $a=1$, $b=4$, $c=13$.
$$
\Delta = 4^2 - 4\cdot1\cdot13 = 16 - 52 = -36 < 0
$$
So there are no real solutions.
Quadratic Equations and Graphs (Connection Only)
The graph of a quadratic function $y = ax^2 + bx + c$ is a parabola. When we solve the quadratic equation
$$
ax^2 + bx + c = 0
$$
we are finding the $x$-values where this parabola crosses (or touches) the $x$-axis, that is, where $y=0$.
- If there are two real solutions, the parabola crosses the $x$-axis at two points.
- If there is one real solution (discriminant $=0$), the parabola just touches the $x$-axis at one point (its vertex).
- If there are no real solutions, the parabola does not touch or cross the $x$-axis at all.
The geometric meaning will be developed more fully in graphing and function chapters; here it is enough to know that solutions to the equation correspond to $x$-intercepts of its graph.
Transforming Problems into Quadratic Equations
Many word problems and formulas lead naturally to quadratic equations once you translate them into algebra. The goal is to recognize when a problem becomes quadratic and to write it in standard form.
Typical patterns that produce quadratic equations include:
- Products involving the same unknown twice, such as $x(x + 5)$, $t(t-3)$, or $w^2$.
- Areas involving length and width that both depend on the same variable.
- Situations involving squares of quantities, such as “the square of a number” or “distance formula” in coordinate geometry.
Basic strategy:
- Define a variable for the unknown quantity.
- Write an equation based on the relationships in the problem.
- Rearrange the equation into the form $ax^2 + bx + c = 0$.
- Recognize that you now have a quadratic equation to solve by some method.
The choice of solving method (factoring, completing the square, quadratic formula) will be discussed in the dedicated chapters that follow.
Summary of Key Points
- A quadratic equation in one variable is any equation that can be written as $ax^2 + bx + c = 0$ with $a \neq 0$.
- The standard form lists terms in order: $x^2$ term, $x$ term, constant term.
- Quadratics can be complete (all three terms) or incomplete (missing $b$ or $c$).
- Solutions (roots) are the $x$-values that make the equation true; a quadratic typically has two, one, or no real solutions.
- The discriminant $\Delta = b^2 - 4ac$ tells you how many real solutions there are, based on its sign.
- Graphically, solving a quadratic equation finds where its parabola crosses or touches the $x$-axis.
- Many real and word problems can be translated into quadratic equations, which are then solved using the specific methods studied in later sections.