Quadratic Equations – Curved Paths and Two Solutions

A quadratic equation contains a variable squared as its highest power. Quadratics appear in physics, engineering, and design — whenever something moves in a curve or an area is involved.

Methods for solving quadratic equations are among the oldest surviving pieces of mathematics — Babylonian clay tablets from around 2000 BCE already contain procedures equivalent to completing the square, and the quadratic formula in essentially its modern form was documented by the Indian mathematician Brahmagupta in 628 CE. The word “quadratic” comes from the Latin quadratus, meaning “square,” because these equations were originally understood geometrically as problems about the area of a square. Every object thrown or launched under gravity — a ball, a fired projectile, a jet of water from a fountain — follows a parabolic (quadratic) path, which is exactly why the discriminant and the formula are so central to physics.

The Standard Form

A quadratic equation has the form ax squared + bx + c = 0, where a is not zero. The highest power of the variable is 2.

Method 1 – Factorising

Write the quadratic as a product of two brackets, then set each bracket equal to zero.

Solve x squared + 5x + 6 = 0.

Find two numbers that multiply to 6 and add to 5: they are 2 and 3. So (x + 2)(x + 3) = 0. Either x + 2 = 0 (x = -2) or x + 3 = 0 (x = -3). Answer: x = -2 or x = -3.

Solve x squared - x - 12 = 0.

Need numbers that multiply to -12 and add to -1: they are -4 and 3. (x - 4)(x + 3) = 0. Answer: x = 4 or x = -3.

Method 2 – Quadratic Formula

When the equation cannot be factorised easily, use the formula. For ax squared + bx + c = 0:

x = (-b plus or minus the square root of (b squared - 4ac)) divided by 2a

Solve 2x squared - 5x + 2 = 0 using the formula.

a=2, b=-5, c=2. Discriminant = 25 - 16 = 9. Square root of 9 = 3. x = (5 plus or minus 3) / 4. x = 8/4 = 2 or x = 2/4 = 0.5. Answer: x = 2 or x = 0.5.

The Discriminant

Value of b squared - 4ac	Number of Solutions	What it Means
Greater than 0	2 real solutions	Parabola crosses the x-axis twice
Equal to 0	1 real solution	Parabola just touches the x-axis
Less than 0	No real solutions	Parabola does not cross the x-axis

Real-Life Application

A ball is thrown upward. Its height in metres after t seconds is h = -5t squared + 20t. When does it hit the ground?

Solve -5t squared + 20t = 0.

Factorise: -5t(t - 4) = 0. t = 0 (launch) or t = 4 (lands). The ball hits the ground after 4 seconds.

Common Mistakes

Forgetting that ax squared + bx + c = 0 requires zero on the right side before solving.
Sign errors when a or b are negative in the quadratic formula.
Only writing one solution when two are possible.

Key Takeaways

Standard form: ax squared + bx + c = 0. Rearrange to this form first.
Try factorising first; use the quadratic formula when factorising is not obvious.
The discriminant tells you how many real solutions exist.

Practice: Quadratic Equations

Solve by Factorising