Quadratic Equations: Key Concepts, Examples, and Study Guide
A quadratic equation can be written as ax² + bx + c = 0, where a ≠ 0. Choose factoring when factors are easy to see, square roots when the linear term is missing, completing the square when a perfect square is convenient, or the quadratic formula in every case. Use the discriminant to predict the roots, then check each result in the original equation and its context.
What Is a Quadratic Equation?
A quadratic equation is an equation whose highest variable power is 2. Its standard form is
ax² + bx + c = 0where a, b, and c are constants and a cannot equal zero. If a = 0, the squared term disappears and the equation becomes linear.
The equation and the related function answer different questions:
- ax² + bx + c = 0 asks which x-values make the expression zero.
- f(x) = ax² + bx + c assigns an output to every allowed input and draws a parabola.
A real solution of the equation is a zero of the function and an x-intercept of the parabola. The functions guide explains the input-output language behind this connection. OpenStax College Algebra 2e presents the same standard form and four solving methods.
Standard, Factored, and Vertex Forms
The same quadratic can appear in several useful forms:
Form | Expression | What it reveals |
Standard | ax² + bx + c | Coefficients for the quadratic formula and discriminant |
Factored | a(x - r₁)(x - r₂) | Real roots r₁ and r₂, when the factorization exists over the reals |
Vertex | a(x - h)² + k | Vertex (h, k) and opening direction |
Changing form does not change the function. It changes which information is easiest to read. Standard form supports the formula, factored form supports the zero-product property, and vertex form supports graph analysis.
How to Choose a Solving Method
Start by setting the equation equal to zero and simplifying. Then use the shape of the equation:
Equation shape | Best first method | Example |
A common factor or clean integer factors | Factoring | x² - 5x + 6 = 0 |
No linear term after rearranging | Square-root property | (x - 4)² = 9 |
A perfect-square structure is close | Completing the square | x² + 6x - 7 = 0 |
Factors are unclear or coefficients are awkward | Quadratic formula | 2x² + 3x - 4 = 0 |
The quadratic formula always works for a genuine quadratic, so it is the reliable fallback. The other methods can be shorter and can reveal structure more clearly. The published solve a quadratic step by step guide carries the extended procedure; this page focuses on choosing and understanding the methods.
Solve by Factoring
Factoring rewrites the polynomial as a product and applies the zero-product property. Solve:
x² - 5x + 6 = 0Find two numbers with product 6 and sum -5: -2 and -3. Then
(x - 2)(x - 3) = 0A product is zero when at least one factor is zero, so x = 2 or x = 3. Check:
- 2² - 5(2) + 6 = 0.
- 3² - 5(3) + 6 = 0.
Factoring is a polynomial skill. The polynomials guide explains factors, zeros, and graph behavior, while factor a quadratic gives the full trinomial-factor process.
Use the Square-Root Property
When a squared expression is isolated, take square roots of both sides and include both signs. Solve:
(x - 4)² = 9Take square roots:
x - 4 = ±3Therefore x = 7 or x = 1. Both values check because (7 - 4)² = 9 and (1 - 4)² = 9.
The ± is essential. Squaring 3 and -3 produces the same result, so a positive square on the right usually gives two real candidates.
Solve by Completing the Square
Completing the square rewrites a quadratic as a squared binomial. Solve:
x² + 6x - 7 = 0Move the constant:
x² + 6x = 7Take half of the x-coefficient and square it: (6/2)² = 9. Add 9 to both sides:
x² + 6x + 9 = 16Rewrite and take square roots:
(x + 3)² = 16x + 3 = ±4So x = 1 or x = -7. Check in the original equation:
- 1² + 6(1) - 7 = 0.
- (-7)² + 6(-7) - 7 = 49 - 42 - 7 = 0.
When a ≠ 1, divide the whole equation by a before completing the square. Add the same value to both sides to preserve equality.
Solve with the Quadratic Formula
For ax² + bx + c = 0, the quadratic formula is
x = (-b ± √(b² - 4ac)) / (2a)Solve 2x² + 3x - 4 = 0. Here a = 2, b = 3, and c = -4:
x = (-3 ± √(3² - 4(2)(-4))) / 4 = (-3 ± √41) / 4These are exact roots. A decimal calculator can approximate them, but keep the radical form when an exact answer is required.
Common formula errors include dropping the negative on -b, forgetting that b itself may be negative, or dividing only the square-root term by 2a. Treat the entire numerator -b ± √(b² - 4ac) as one quantity.
Use the Discriminant Before You Solve
The expression under the radical,
D = b² - 4acis the discriminant. It predicts the root type:
- D > 0: two distinct real roots.
- D = 0: one repeated real root.
- D < 0: two complex roots and no real x-intercepts.
Calculate D before using the full formula. It tells you how many real answers to expect and helps catch arithmetic mistakes. For x² + 4x + 8 = 0, D = 16 - 32 = -16, so a pair of real roots would signal an error.
A negative discriminant means no real roots, not “no answer.” Over the complex numbers, the equation still has two roots.
Roots, Zeros, and the Parabola
The words root, solution, and zero all point to an x-value that makes the quadratic expression equal zero. On the graph of y = ax² + bx + c, each real root is an x-intercept.
For x² - 2x - 3 = 0, factoring gives (x - 3)(x + 1) = 0, so the roots are 3 and -1. The parabola crosses the x-axis at (3, 0) and (-1, 0).
If D = 0, the parabola touches the x-axis once at the vertex. If D < 0, it never reaches the x-axis. The functions guide develops vertex, domain, range, and graph-reading ideas in a wider setting.
Apply Quadratics and Filter the Roots
A quadratic model can produce algebraically valid roots that do not fit the situation. Suppose a rectangle has width x, length x + 3, and area 40:
x(x + 3) = 40x² + 3x - 40 = 0Factor:
(x + 8)(x - 5) = 0The algebraic roots are x = -8 and x = 5. A physical width cannot be negative, so the model keeps x = 5. The rectangle is 5 by 8, and 5 × 8 = 40 confirms the result.
Always separate two checks: first, does the value solve the original equation? Second, does it satisfy the restrictions of the problem?
Check Every Solution
Use this routine after any method:
- Substitute every candidate into the original equation.
- Evaluate both sides independently.
- Confirm the values match.
- Apply domain and context restrictions.
- Compare the number and type of roots with the discriminant.
The last comparison is especially useful. A positive discriminant should not end with one real root, and a negative discriminant should not produce a real x-intercept.
Common Mistakes
- Forgetting to set the equation equal to zero before factoring.
- Letting a = 0 and still calling the equation quadratic.
- Dropping the ± after taking square roots.
- Using the wrong signed value for b or c in the formula.
- Adding the completing-square value to only one side.
- Dividing only part of the quadratic-formula numerator by 2a.
- Calling complex roots real x-intercepts.
- Keeping a negative time, length, or count that the context rejects.
- Checking only one candidate root.
Where Quadratic Equations Fit in Algebra
Quadratics connect polynomial operations, equation solving, and parabolas. The Algebra 1 guide covers the foundations, while the Algebra 2 guide extends quadratics to complex roots, transformations, and broader function work.
A parabola can also be paired with a line or another curve. The systems guide introduces simultaneous conditions before you move to nonlinear systems.
How to Practice Quadratic Equations with Gauth
Choose one method or one diagnostic skill at a time. Open Gauth Math Courses, select Create Course, and enter a topic such as "completing the square and checking quadratic roots" plus your level. Complete one activity, then solve a fresh equation on paper and compare the roots with the discriminant.
Sources and Review
- OpenStax College Algebra 2e: Quadratic Equations, accessed October 13, 2026.