Concept Overview
A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \), where \( a \ne 0 \). Quadratics appear in motion, geometry, economics, and more. This page explores several methods to solve quadratic equations and interpret their solutions.
Key Vocabulary
- Quadratic equation: An equation in the form \( ax^2 + bx + c = 0 \).
- Parabola: The U-shaped graph of a quadratic.
- Coefficient: The numbers \( a, b, \) and \( c \) in the standard form.
- Leading coefficient: The coefficient \( a \) of the \( x^2 \) term, which determines the direction of the parabola’s opening.
- Factoring: Rewriting the quadratic as a product of two binomials.
- Completing the square: A method to solve quadratics by rewriting them as a perfect square trinomial.
- Root (or solution): A value of \( x \) that makes the equation true.
- Vertex: The turning point of the graph of a quadratic.
- Discriminant: The expression \( b^2 – 4ac \), which tells how many real roots exist.
- Axis of symmetry: A vertical line that divides the parabola into mirror halves.
Graphing \( y = x^2 \)
\( y = x^2 \) It is the most basic example of a quadratic function, where the output is equal to the square of the input.
When values for \( x \) are substituted into the expression \( x^2 \), the corresponding values of \( y \) are always positive or zero, and they grow more quickly as \( x \) moves away from zero.
The result is a curved shape called a parabola. For \( y = x^2 \), this parabola opens upward and is symmetric about the vertical line \( x = 0 \).
Table of Values
| \( x \) | \( y = x^2 \) |
|---|---|
| \( -3 \) | \( 9 \) |
| \( -2 \) | \( 4 \) |
| \( -1 \) | \( 1 \) |
| \( 0 \) | \( 0 \) |
| \( 1 \) | \( 1 \) |
| \( 2 \) | \( 4 \) |
| \( 3 \) | \( 9 \) |
As shown in the table, each positive and negative pair of \( x \)-values produces the same \( y \)-value, which is why the graph is symmetric.
This simple curve forms the foundation for understanding all other quadratic graphs. Any equation in the form \( y = ax^2 + bx + c \) is a variation of this shape — it may be stretched, shifted, or flipped, but the core structure remains the same.
Standard Form and Properties
A quadratic equation in standard form is written as:
Standard Form
\( ax^2 + bx + c = 0 \)
where \( a, b, \) and \( c \) are real numbers, and \( a \ne 0 \).
This form is useful for solving equations, analyzing graphs, and applying methods like factoring or the quadratic formula. Each coefficient reveals something about the shape or position of the graph.
Direction of Opening
If \( a > 0 \), the parabola opens upward.
If \( a < 0 \), the parabola opens downward.
The sign of \( a \) tells us whether the graph is concave up or down.
Axis of Symmetry
\( x = \frac{-b}{2a} \)
This vertical line passes through the vertex and splits the parabola into mirror-image halves.
Vertex of a Parabola
\( \left( \frac{-b}{2a},\ \frac{-D}{4a} \right) \)
For a quadratic function written in standard form: \[ y = ax^2 + bx + c \] the graph is a parabola. Its vertex is the point at which the graph reaches its minimum (if \( a > 0 \)) or maximum (if \( a < 0 \)).
The x-coordinate of the vertex is: \[ x = \frac{-b}{2a} \] This value also defines the axis of symmetry.
To find the y-coordinate of the vertex, substitute \( x = \frac{-b}{2a} \) into the equation, or use this equivalent expression: \[ y = \frac{-D}{4a}, \quad \text{where } D = b^2 – 4ac \]
Equivalent forms: \[ \frac{-D}{4a} = \frac{-(b^2 – 4ac)}{4a} = \frac{4ac – b^2}{4a} = f\left(\frac{-b}{2a}\right) \]
These forms all represent the same y-value at the vertex. They arise by evaluating the original function at the axis of symmetry: \[ y = a\left( \frac{-b}{2a} \right)^2 + b\left( \frac{-b}{2a} \right) + c \]
y-Intercept
\( c \)
The constant term \( c \) gives the point where the parabola intersects the y-axis, at \( (0, c) \).
Vertex Form of a Quadratic
A quadratic expression can also be written in vertex form:
Vertex Form
\( y = a(x – h)^2 + k \)
where \( (h, k) \) is the vertex of the parabola and \( a \) determines the direction and width of opening.
Vertex form is especially helpful for graphing, because it shows at a glance:
- The vertex is \( (h, k) \)
- The axis of symmetry is \( x = h \)
- The parabola opens upward if \( a > 0 \), and downward if \( a < 0 \)
Example
Given: \( y = 2(x + 1)^2 – 5 \)
- Vertex: \( (-1, -5) \)
- Axis of symmetry: \( x = -1 \)
- Opens upward (since \( a = 2 > 0 \))
What Does It Mean to Solve a Quadratic Equation?
Solving a quadratic equation means finding the values of \( x \) that make the equation true — in other words, the values that satisfy \[ ax^2 + bx + c = 0 \] These values are called the solutions, roots, or zeros of the equation.
Algebraically, a solution is any number that can be substituted for \( x \) to make the left side equal to zero. Graphically, the solutions correspond to the x-intercepts of the parabola \( y = ax^2 + bx + c \).
Key Idea
To solve: Find all values of \( x \) where \( ax^2 + bx + c = 0 \)
Some equations have two real solutions, some have one repeated solution, and others have no real solutions at all. The structure of the quadratic determines which of these is true.
Solving by Graphing
Quadratic equations can be solved by graphing the related formula: \[ y = ax^2 + bx + c \] The solutions (or roots) of the equation \( ax^2 + bx + c = 0 \) are the x-values where the graph intersects the x-axis. These are called the x-intercepts.
Graphing Insight
The solutions of \( ax^2 + bx + c = 0 \) appear as the points where the curve touches or crosses the x-axis.
Example 1 (Two real solutions)
Solve: \( x^2 – 4 = 0 \) by graphing
\( y = x^2 – 4 \) is a parabola that opens upward and crosses the x-axis at \( x = -2 \) and \( x = 2 \).
Example 2 (One solution)
Solve: \( x^2 + 4x + 4 = 0 \) by graphing
\( y = (x + 2)^2 \) is a parabola with vertex at \( (-2, 0) \). It touches the x-axis at exactly one point.
Example 3 (No real solution)
Solve: \( x^2 + 2x + 5 = 0 \) by graphing
\( y = x^2 + 2x + 5 \) is a parabola that opens upward and never intersects the x-axis.
This parabola has no real roots.
Graphing is a useful visual method for estimating or verifying solutions. When exact roots are irrational or complex, graphing provides approximate locations and insight into the nature of the solutions.
Solving by Factoring
When a quadratic equation is written in standard form, one method of solving it is by factoring the left-hand side and applying the Zero Product Property.
Zero Product Property
If \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \)
The overall strategy is:
- Write the equation in standard form: \( ax^2 + bx + c = 0 \)
- Factor the quadratic expression on the left
- Set each factor equal to zero
- Solve each resulting equation
Example 1 (Simple trinomial)
Solve: \( x^2 + 5x + 6 = 0 \)
Factor: \( (x + 2)(x + 3) = 0 \)
Apply zero product rule: \( x + 2 = 0 \) or \( x + 3 = 0 \)
Solutions: \( x = -2 \), \( x = -3 \)
Example 2 (With GCF)
Solve: \( 3x^2 – 6x = 0 \)
Factor out GCF: \( 3x(x – 2) = 0 \)
Set factors to zero: \( 3x = 0 \) or \( x – 2 = 0 \)
Solutions: \( x = 0 \), \( x = 2 \)
Example 3 (General trinomial)
Solve: \( 2x^2 + 5x + 3 = 0 \)
Factor: \( (2x + 3)(x + 1) = 0 \)
Set each factor to zero: \( 2x + 3 = 0 \), \( x + 1 = 0 \)
Solutions: \( x = -\frac{3}{2} \), \( x = -1 \)
If a quadratic expression cannot be factored easily with rational numbers, other methods like completing the square or the quadratic formula are preferred. But when factoring works, it is often the most direct approach.
Review: Simplifying Square Roots
When solving quadratics using the quadratic formula or completing the square, you’ll often encounter square roots. These need to be simplified to express solutions in their simplest form.
Perfect Square Factors
\( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \)
Use this to break down radicals by factoring out perfect squares.
Example 1
Simplify: \( \sqrt{18} \)
- Factor: \( \sqrt{18} = \sqrt{9 \cdot 2} \)
- Apply rule: \( \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \)
Example 2
Simplify: \( \frac{\sqrt{50}}{2} \)
- Factor: \( \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2} \)
- Simplified fraction: \( \frac{5\sqrt{2}}{2} \)
Example 3
Simplify: \( \frac{-3 \pm \sqrt{48}}{6} \)
- Factor: \( \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} \)
- Substitute: \( \frac{-3 \pm 4\sqrt{3}}{6} \)
- Break up the fraction: \( \frac{-3}{6} \pm \frac{4\sqrt{3}}{6} \)
- Simplify each part: \( -\frac{1}{2} \pm \frac{2\sqrt{3}}{3} \)
- Final form: \( x = -\frac{1}{2} \pm \frac{2\sqrt{3}}{3} \)
This technique will help you express irrational solutions in their most concise form. It’s also useful for comparing, estimating, or checking roots on a calculator.
Solving by Completing the Square
Completing the square is a method of solving quadratic equations by rewriting the expression as a perfect square trinomial. This form allows the equation to be solved by taking square roots instead of factoring. It is also the foundation for converting to vertex form.
Core Idea
\( x^2 + bx \rightarrow x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2 \)
The perfect square trinomial is created by adding the square of half the linear coefficient. This produces an equivalent expression in the form of a binomial squared.
Steps for solving \( ax^2 + bx + c = 0 \) by completing the square:
- Move the constant term to the other side
- If \( a \ne 1 \), divide the entire equation by \( a \)
- Add \( \left(\frac{b}{2}\right)^2 \) to both sides to complete the square
- Rewrite the left side as a squared binomial
- Take the square root of both sides
- Simplify to isolate \( x \)
Example 1 (Simple case)
Solve: \( x^2 + 6x + 5 = 0 \)
- Move constant: \( x^2 + 6x = -5 \)
- Complete the square: \( x^2 + 6x + 9 = -5 + 9 \)
- \( (x + 3)^2 = 4 \)
- \( x + 3 = \pm 2 \)
- So \( x = -3 + 2 = -1 \) and \( x = -3 – 2 = -5 \)
- Solutions: \( x = -1 \), \( x = -5 \)
Example 2 (With leading coefficient)
Solve: \( 2x^2 + 8x – 6 = 0 \)
- Move constant: \( 2x^2 + 8x = 6 \)
- Divide by 2: \( x^2 + 4x = 3 \)
- Complete the square: \( x^2 + 4x + 4 = 3 + 4 \)
- \( (x + 2)^2 = 7 \)
- \( x = -2 \pm \sqrt{7} \)
Example 3 (Fractional result)
Solve: \( x^2 – 5x + 1 = 0 \)
- Move constant: \( x^2 – 5x = -1 \)
- Complete the square: \( x^2 – 5x + \frac{25}{4} = -1 + \frac{25}{4} \)
- \( \left(x – \frac{5}{2}\right)^2 = \frac{21}{4} \)
- \( x = \frac{5}{2} \pm \frac{\sqrt{21}}{2} \)
- Complete the square: \( y = (x^2 + 6x + 9) + 5 – 9 \) (Add and subtract 9 to keep the equation balanced; 9 is \( \left(\frac{6}{2}\right)^2 \))
- Vertex form: \( y = (x + 3)^2 – 4 \)
- Vertex: \( (-3, -4) \)
This method works for any quadratic equation, regardless of whether it can be factored. It is especially helpful for equations with irrational or complex solutions.
Solving with the Quadratic Formula
The quadratic formula is a universal method for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It works whether the expression can be factored or not, and whether the roots are rational, irrational, or complex.
Quadratic Formula
\( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \)
This formula gives both solutions (“roots”) of the quadratic. The expression under the square root, \( b^2 – 4ac \), is called the discriminant.
Steps:
- Identify the coefficients \( a \), \( b \), and \( c \) from the equation in standard form
- Substitute into the quadratic formula
- Simplify under the square root (the discriminant)
- Simplify the entire expression to find the two values of \( x \)
Example 1 (Rational roots)
Solve: \( x^2 – 5x + 6 = 0 \)
- \( a = 1 \), \( b = -5 \), \( c = 6 \)
- \( x = \frac{-(-5) \pm \sqrt{(-5)^2 – 4(1)(6)}}{2(1)} \)
- \( x = \frac{5 \pm \sqrt{25 – 24}}{2} = \frac{5 \pm 1}{2} \)
- Solutions: \( x = 3 \), \( x = 2 \)
Example 2 (Irrational roots)
Solve: \( 2x^2 + 3x – 2 = 0 \)
- \( a = 2 \), \( b = 3 \), \( c = -2 \)
- \( x = \frac{-3 \pm \sqrt{3^2 – 4(2)(-2)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 16}}{4} \)
- \( x = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4} \)
- Solutions: \( x = \frac{1}{2},\ x = -2 \)
Example 3 (No real solution)
Solve: \( x^2 + 2x + 5 = 0 \)
- \( a = 1 \), \( b = 2 \), \( c = 5 \)
- \( x = \frac{-2 \pm \sqrt{2^2 – 4(1)(5)}}{2} = \frac{-2 \pm \sqrt{4 – 20}}{2} \)
- \( x = \frac{-2 \pm \sqrt{-16}}{2} \Rightarrow \text{no real solutions} \)
Using the Discriminant
The discriminant is the part of the quadratic formula under the square root:
Discriminant
\( D = b^2 – 4ac \)
The value of \( D \) determines how many real solutions a quadratic equation has, and what type they are.
The discriminant gives a quick way to understand the structure of a quadratic equation without solving it fully.
Discriminant Cases
Case 1: \( D > 0 \) → two distinct real solutions
Case 2: \( D = 0 \) → one repeated real solution
Case 3: \( D < 0 \) → no real solutions (complex roots)
Example 1 (Two real solutions)
Equation: \( x^2 – 5x + 6 = 0 \)
Discriminant: \( D = (-5)^2 – 4(1)(6) = 25 – 24 = 1 \)
Since \( D > 0 \), there are two distinct real solutions.
Example 2 (One real solution)
Equation: \( x^2 + 6x + 9 = 0 \)
\( D = (6)^2 – 4(1)(9) = 36 – 36 = 0 \)
One repeated real root at \( x = -3 \)
Example 3 (No real solutions)
Equation: \( 2x^2 + x + 5 = 0 \)
\( D = (1)^2 – 4(2)(5) = 1 – 40 = -39 \)
Since \( D < 0 \), there are no real solutions; the roots are complex.
The discriminant is a diagnostic tool: it doesn’t solve the equation by itself, but it tells what kind of solution to expect. This can help decide which method is most efficient or whether real solutions even exist.
Try It Yourself!
- Solve: \( x^2 – 5x + 6 = 0 \) by factoring
- Estimate roots of \( x^2 + 2x – 3 \) by graphing (e.g., by hand or using a calculator)
- Rewrite \( x^2 + 4x + 1 = 0 \) in vertex form by completing the square
- Determine the number of real solutions to \( 4x^2 + 2x + 5 = 0 \)
- \( (x – 2)(x – 3) \Rightarrow x = 2, 3 \)
- \( (x + 2)^2 = 3 \Rightarrow x = -2 \pm \sqrt{3} \)
- \( x = \frac{1 \pm \sqrt{25}}{6} = \frac{1 \pm 5}{6} \Rightarrow x = 1 \) and \( x = -\frac{2}{3} \)
- Roots at \( x = -3, 1 \)
- Discriminant \( b^2 – 4ac = 4 – 80 = -76 \Rightarrow \) no real solutions
Reveal Answers
Connections & Extensions
Quadratic equations appear in physics (projectile motion), finance (optimization), and geometry (area models).