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.
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.
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.
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
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?
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 Questions
- Solve x squared - 7x + 10 = 0 by factorising.
- Solve x squared + 2x - 8 = 0 by factorising.
- Solve 3x squared - 10x + 3 = 0 using the formula.
- How many solutions does 2x squared + x + 5 = 0 have? Justify using the discriminant.
- A rectangle has length (x + 3) and width (x - 1) and area 28. Find x.