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Quadratic Equation: ax²+bx+c=0, Formula & Examples

Quadratic Equation

An equation includes only second degree polynomials. Some examples are y = 3x2 – 5x2 + 1, x2 + 5xy + y2 = 1, and 1.6a2 +5.9a – 3.14 = 0.

Note: When there is only one variable, a quadratic equation can be expressed in the form ax2 + bx + c = 0 where a, b, and c are all constants.

See also

Quadratic formula

Key Formula

ax2+bx+c=0⟹x=−b±b2−4ac2aax^2 + bx + c = 0 \quad \Longrightarrow \quad x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Where:
  • aa = The coefficient of x², must not equal zero
  • bb = The coefficient of x (the first-degree term)
  • cc = The constant term (no variable attached)
  • xx = The unknown variable you solve for
  • b2−4acb^2 - 4ac = The discriminant, which determines the number and type of solutions

Worked Example

Problem: Solve the quadratic equation x² + 6x + 5 = 0.
Step 1: Identify the coefficients a, b, and c from the standard form ax² + bx + c = 0.
a=1,b=6,c=5a = 1, \quad b = 6, \quad c = 5
Step 2: Calculate the discriminant to understand what kind of solutions to expect.
b2−4ac=62−4(1)(5)=36−20=16b^2 - 4ac = 6^2 - 4(1)(5) = 36 - 20 = 16
Step 3: Since the discriminant is positive, there are two distinct real solutions. Apply the quadratic formula.
x=−6±162(1)=−6±42x = \frac{-6 \pm \sqrt{16}}{2(1)} = \frac{-6 \pm 4}{2}
Step 4: Compute the two solutions by using + and − separately.
x=−6+42=−22=−1orx=−6−42=−102=−5x = \frac{-6 + 4}{2} = \frac{-2}{2} = -1 \qquad \text{or} \qquad x = \frac{-6 - 4}{2} = \frac{-10}{2} = -5
Step 5: Verify by factoring: x² + 6x + 5 = (x + 1)(x + 5) = 0, which gives x = −1 or x = −5. ✓
(x+1)(x+5)=0(x + 1)(x + 5) = 0
Answer: x = −1 or x = −5

Another Example

This example demonstrates the special case where the discriminant equals zero, producing exactly one repeated solution instead of two distinct solutions.

Problem: Solve the quadratic equation 2x² − 4x + 2 = 0.
Step 1: Identify the coefficients from the standard form.
a=2,b=−4,c=2a = 2, \quad b = -4, \quad c = 2
Step 2: Calculate the discriminant.
b2−4ac=(−4)2−4(2)(2)=16−16=0b^2 - 4ac = (-4)^2 - 4(2)(2) = 16 - 16 = 0
Step 3: A discriminant of zero means the equation has exactly one repeated (double) solution. Apply the quadratic formula.
x=−(−4)±02(2)=44=1x = \frac{-(-4) \pm \sqrt{0}}{2(2)} = \frac{4}{4} = 1
Step 4: Verify: 2(1)² − 4(1) + 2 = 2 − 4 + 2 = 0. ✓ The parabola touches the x-axis at exactly one point.
2(1)2−4(1)+2=02(1)^2 - 4(1) + 2 = 0
Answer: x = 1 (a repeated root)

Frequently Asked Questions

How many solutions can a quadratic equation have?
A quadratic equation in one variable can have two distinct real solutions, one repeated real solution, or no real solutions (two complex solutions). The discriminant b2−4acb^2 - 4ac determines which case applies: positive gives two real solutions, zero gives one repeated solution, and negative gives two complex solutions.
What is the difference between a quadratic equation and a quadratic expression?
A quadratic expression is a polynomial of degree two, such as 3x2+2x−73x^2 + 2x - 7. A quadratic equation sets that expression equal to something, typically zero: 3x2+2x−7=03x^2 + 2x - 7 = 0. The equation asks you to find specific values of xx; the expression is just a mathematical phrase with no equals sign requiring a solution.
When should you use the quadratic formula instead of factoring?
Factoring is often faster when the coefficients are small integers and the roots are rational. However, many quadratic equations do not factor neatly. The quadratic formula works on every quadratic equation regardless of the coefficients, so use it whenever factoring is not obvious or the discriminant is not a perfect square.

Quadratic Equation vs. Linear Equation

Quadratic EquationLinear Equation
DegreeDegree 2 (highest power of the variable is 2)Degree 1 (highest power of the variable is 1)
Standard formax² + bx + c = 0ax + b = 0
Number of solutionsUp to 2 real solutionsExactly 1 solution (if a ≠ 0)
Graph shapeParabola (U-shaped curve)Straight line
Solving methodsFactoring, completing the square, quadratic formulaIsolate the variable using inverse operations

Why It Matters

Quadratic equations appear throughout algebra, physics, and engineering. In physics, projectile motion is modeled by quadratic equations — for instance, finding when a ball hits the ground requires solving one. They are central to algebra courses from middle school through college and form the foundation for studying higher-degree polynomials and conic sections.

Common Mistakes

Mistake: Forgetting that a ≠ 0 and treating a linear equation as quadratic.
Correction: If the coefficient of x² is zero, the equation reduces to a linear equation (degree 1), not a quadratic. Always check that a ≠ 0 before applying quadratic methods.
Mistake: Dropping the ± sign in the quadratic formula and finding only one solution.
Correction: The ± symbol means you must compute two separate values: one using + and one using −. Both are potential solutions. Only when the discriminant is zero do the two values coincide.

Related Terms

  • Quadratic Formula — The formula used to solve any quadratic equation
  • Equation — General concept that a quadratic equation is a type of
  • Degree of a Polynomial — A quadratic has degree two
  • Polynomial — A quadratic is a second-degree polynomial
  • Variable — The unknown quantity solved for in the equation
  • Constant — The coefficients a, b, and c are constants
  • Discriminant — Determines the number and type of solutions
  • Factoring — An alternative method for solving quadratic equations