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Quadratic

Quadratic

An equation, graph, or data that can be modeled by a degree 2 polynomial.

 

 

See also

Quadratic polynomial, quadratic equation, quadratic formula

Key Formula

ax2+bx+c=0ax^2 + bx + c = 0
Where:
  • aa = The coefficient of the squared term; must not equal 0 (otherwise the expression is linear, not quadratic)
  • bb = The coefficient of the first-degree (linear) term
  • cc = The constant term
  • xx = The variable being solved for

Worked Example

Problem: Solve the quadratic equation x2−5x+6=0x^2 - 5x + 6 = 0.
Step 1: Identify the coefficients: a=1a = 1, b=−5b = -5, c=6c = 6.
x2−5x+6=0x^2 - 5x + 6 = 0
Step 2: Factor the quadratic. Look for two numbers that multiply to 6 and add to −5-5. Those numbers are −2-2 and −3-3.
(x−2)(x−3)=0(x - 2)(x - 3) = 0
Step 3: Set each factor equal to zero and solve.
x−2=0⇒x=2x - 2 = 0 \quad \Rightarrow \quad x = 2
Step 4: Solve the second factor.
x−3=0⇒x=3x - 3 = 0 \quad \Rightarrow \quad x = 3
Answer: The solutions are x=2x = 2 and x=3x = 3.

Another Example

This example applies a quadratic to a real-world projectile motion scenario, showing that quadratic equations model physical situations — not just abstract algebra. It also demonstrates that one solution (t=0t = 0) can be valid mathematically but irrelevant in context.

Problem: A ball is launched upward from the ground with an initial velocity of 20 m/s. Its height in meters after tt seconds is given by h=−5t2+20th = -5t^2 + 20t. When does the ball hit the ground again?
Step 1: The ball hits the ground when h=0h = 0. Set the quadratic equal to zero.
−5t2+20t=0-5t^2 + 20t = 0
Step 2: Factor out the greatest common factor, −5t-5t.
−5t(t−4)=0-5t(t - 4) = 0
Step 3: Set each factor equal to zero: −5t=0-5t = 0 gives t=0t = 0 (the launch time), and t−4=0t - 4 = 0 gives t=4t = 4.
t=0ort=4t = 0 \quad \text{or} \quad t = 4
Answer: The ball hits the ground again at t=4t = 4 seconds.

Frequently Asked Questions

What is the difference between a quadratic equation and the quadratic formula?
A quadratic equation is any equation of the form ax2+bx+c=0ax^2 + bx + c = 0. The quadratic formula, x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, is a specific method for solving any quadratic equation. The equation is the problem; the formula is one tool for finding the answer.
Why is it called 'quadratic' if the highest power is 2, not 4?
The word comes from the Latin "quadratus," meaning "square." A square with side length xx has area x2x^2, so expressions involving x2x^2 became known as quadratic. It refers to squaring, not to the number four.
How many solutions can a quadratic equation have?
A quadratic equation 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.

Quadratic (degree 2) vs. Linear (degree 1)

Quadratic (degree 2)Linear (degree 1)
General formax2+bx+c=0ax^2 + bx + c = 0mx+b=0mx + b = 0
Highest power of variable21
Graph shapeParabola (U-shaped curve)Straight line
Maximum number of real solutions21
Common solving methodsFactoring, quadratic formula, completing the squareIsolate the variable using inverse operations

Why It Matters

Quadratics appear throughout algebra, geometry, and physics — from calculating the area of rectangles to modeling projectile motion and free-fall problems. They are central to high school math courses and standardized tests like the SAT and ACT. Understanding quadratics also builds the foundation for studying higher-degree polynomials and more advanced functions in precalculus and calculus.

Common Mistakes

Mistake: Forgetting that a≠0a \neq 0. Students sometimes write an expression like 0x2+3x+10x^2 + 3x + 1 and call it quadratic.
Correction: If a=0a = 0, the x2x^2 term disappears and the expression becomes linear (3x+13x + 1). A quadratic must have a nonzero coefficient on the x2x^2 term.
Mistake: Assuming every quadratic equation has two distinct real solutions.
Correction: Check the discriminant b2−4acb^2 - 4ac first. When it equals zero, there is exactly one repeated real solution. When it is negative, there are no real solutions — only complex ones.

Related Terms