How to Solve Quadratic Equations?
In this article, we will cover:
- What is a quadratic equation?
- The standard form of a quadratic equation
- Simple quadratic equations
- The number of solutions in a quadratic equation
- Solving quadratic equations by factorisation
- Using the quadratic formula
- Understanding the discriminant
- Solving quadratic equations by completing the square
What is a Quadratic Equation?
A quadratic equation is an algebraic equation where the highest power of the pronumeral, usually x, is 2. When a quadratic relation is plotted on a graph, it forms a Parabola.
The Standard Form of a Quadratic Equation
The standard form of a quadratic equation is:
ax² + bx + c = 0, where a ≠ 0
The value of a cannot be 0 because the ax² term would disappear, and the equation would become linear, not quadratic.
Number of Solutions for a Quadratic Equation
The most simple quadratic equation is in the form of: x² = c.
To solve for x, we take the square root of both sides. Because squaring a negative number gives a positive result, we must account for both positive and negative possibilities: x = +√c or -√c
The number of solutions for x depends entirely on the value of c:
If c is positive (c > 0): x has 2 real solutions. (For example, if x² = 16, then x = 4 or x = -4).
If c is zero (c = 0): x has 1 real solution. x² = 0, then x = 0
If c is negative (c < 0): x has no real solutions. This is because no real number multiplied by itself will give a negative result (you cannot take the square root of a negative number).
| Equation | Number of Solutions | Example |
|---|---|---|
| x² = c, where c > 0 | 2 solutions | x² = 16, hence x = ±4 |
| x² = 0 | 1 solution | x = 0 |
| x² = c, where c < 0 | No real solution | x² = -9; √-9 is not real |
Factorisation Method
The factorisation method is used when the quadratic expression can be factored easily.
Steps:
- Convert the equation into standard form.
- Split the middle term b into two numbers whose sum is b and product is a × c.
- Factor the quadratic.
- Set each factor equal to zero.
- Solve for x.
Example: Solve x² − 5x + 6 = 0
x² − 5x + 6 = 0
Here, a = 1, b = -5 and c = 6.
We need two numbers whose sum is -5 and whose product is 6.
The numbers are -3 and -2.
x² − 5x + 6 = 0
x² − 3x − 2x + 6 = 0
x(x − 3) − 2(x − 3) = 0
(x − 3)(x − 2) = 0
Therefore, either:
x − 3 = 0 or x − 2 = 0
Therefore:
x = 3 or x = 2
The Quadratic Formula
The quadratic formula is a universal method because it can be used to solve any quadratic equation.
The formula is:
x = (-b ± √(b² − 4ac)) / 2a
This means:
x₁ = (-b + √(b² − 4ac)) / 2a
and
x₂ = (-b − √(b² − 4ac)) / 2a
Once you know the roots, the quadratic can be written as:
a(x − x₁)(x − x₂) = 0
The Discriminant in Quadratic Equations
The discriminant tells us how many real solutions a quadratic equation has.
D = b² − 4ac
- If D > 0, there are two real solutions.
- If D = 0, there is one real solution or equal roots.
- If D < 0, there are no real solutions.
Completing the Square
Completing the square is used when a quadratic equation cannot be factorised easily using integers.
Steps:
- Make the coefficient of x² equal to 1.
- Move the constant to the other side.
- Take half of the coefficient of x and square it: (b/2)².
- Add this number to both sides.
- Write the left side as a perfect square.
- Take the square root of both sides and solve for x.
Example: Solve x² + 6x + 5 = 0
Step 1: Move the constant to the other side
x² + 6x = -5
Step 2: Take half of 6 and square it
(6/2)² = 3² = 9
Step 3: Add 9 to both sides
x² + 6x + 9 = -5 + 9
Step 4: Write the left side as a square
(x + 3)² = 4
Step 5: Take the square root of both sides
x + 3 = ±2
Step 6: Solve for x
x = -3 ± 2
Therefore:
x = -1 or x = -5
Final Answer: x = -1 or x = -5
