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Quadratic Formula

Quadratic Formula

A formula for the roots of a quadratic equation.

 

Quadratic equation ax²+bx+c=0 and its roots formula x=(-b±√(b²-4ac))/2a

Key Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Where:
  • xx = The solutions (roots) of the quadratic equation
  • aa = The coefficient of $x^2$ (must not be zero)
  • bb = The coefficient of $x$
  • cc = The constant term
  • b2−4acb^2 - 4ac = The discriminant, which determines the number and type of solutions

Worked Example

Problem: Solve the quadratic equation 2x2+6x−20=02x^2 + 6x - 20 = 0 using the quadratic formula.
Step 1: Identify the coefficients from the equation 2x2+6x−20=02x^2 + 6x - 20 = 0.
a=2,b=6,c=−20a = 2, \quad b = 6, \quad c = -20
Step 2: Substitute the coefficients into the quadratic formula.
x=−6±62−4(2)(−20)2(2)x = \frac{-6 \pm \sqrt{6^2 - 4(2)(-20)}}{2(2)}
Step 3: Compute the discriminant under the square root.
b2−4ac=36−(−160)=36+160=196b^2 - 4ac = 36 - (-160) = 36 + 160 = 196
Step 4: Simplify the square root and compute both solutions.
x=−6±1964=−6±144x = \frac{-6 \pm \sqrt{196}}{4} = \frac{-6 \pm 14}{4}
Step 5: Split into two solutions using ++ and −-.
x=−6+144=84=2orx=−6−144=−204=−5x = \frac{-6 + 14}{4} = \frac{8}{4} = 2 \qquad \text{or} \qquad x = \frac{-6 - 14}{4} = \frac{-20}{4} = -5
Answer: The two solutions are x=2x = 2 and x=−5x = -5.

Another Example

Problem: Solve x2+2x+5=0x^2 + 2x + 5 = 0 using the quadratic formula.
Step 1: Identify the coefficients.
a=1,b=2,c=5a = 1, \quad b = 2, \quad c = 5
Step 2: Substitute into the formula.
x=−2±4−202=−2±−162x = \frac{-2 \pm \sqrt{4 - 20}}{2} = \frac{-2 \pm \sqrt{-16}}{2}
Step 3: The discriminant is negative, so the square root involves an imaginary number.
x=−2±4i2=−1±2ix = \frac{-2 \pm 4i}{2} = -1 \pm 2i
Answer: The two complex solutions are x=−1+2ix = -1 + 2i and x=−1−2ix = -1 - 2i. A negative discriminant means the parabola does not cross the xx-axis.

Frequently Asked Questions

What does the discriminant tell you about the solutions?
The discriminant is the expression b2−4acb^2 - 4ac under the square root. If it is positive, the equation has two distinct real solutions. If it equals zero, there is exactly one real solution (a repeated root). If it is negative, there are no real solutions — only two complex conjugate solutions.
When should you use the quadratic formula instead of factoring?
The quadratic formula works for every quadratic equation, whether or not it factors neatly. Use factoring when the roots are easy integers or simple fractions. Use the quadratic formula when factoring is not obvious, or when you need exact irrational or complex roots.

Quadratic Formula vs. Completing the Square

Both methods solve any quadratic equation. Completing the square rewrites the equation as (x−h)2=k(x - h)^2 = k and then takes a square root. The quadratic formula is actually derived by completing the square on the general equation ax2+bx+c=0ax^2 + bx + c = 0, so it is essentially a shortcut for the same process. Completing the square is also useful for rewriting quadratics in vertex form, while the quadratic formula is specifically designed to find roots quickly.

Why It Matters

The quadratic formula is one of the most widely used formulas in algebra because it guarantees a solution to any quadratic equation, regardless of whether it can be factored. It appears throughout physics (projectile motion, energy equations), engineering, and economics whenever a relationship involves a squared variable. Understanding it also introduces the discriminant, which tells you about the nature of solutions without solving the equation.

Common Mistakes

Mistake: Forgetting to negate bb — writing b±…2a\frac{b \pm \sqrt{\ldots}}{2a} instead of −b±…2a\frac{-b \pm \sqrt{\ldots}}{2a}.
Correction: The formula begins with −b-b. A helpful habit is to say "negative bb" aloud every time you write the numerator.
Mistake: Dividing only part of the numerator by 2a2a — for example, computing −b2a±b2−4ac\frac{-b}{2a} \pm \sqrt{b^2 - 4ac} without dividing the square root term by 2a2a as well.
Correction: The entire numerator −b±b2−4ac-b \pm \sqrt{b^2 - 4ac} is divided by 2a2a. Use a fraction bar that clearly extends under both terms, or add parentheses: (−b±b2−4ac)/(2a)(-b \pm \sqrt{b^2-4ac})/(2a).

Related Terms

  • Quadratic Equation — The type of equation the formula solves
  • Discriminant — The expression b2−4acb^2 - 4ac inside the formula
  • Root — The solutions produced by the formula
  • Factoring — An alternative method for solving quadratics
  • Completing the Square — The method used to derive the formula
  • Parabola — The graph whose x-intercepts the formula finds
  • Formula — General term for a mathematical relationship