In one line
To solve x² + 5x + 6 = 0, find the values of x that make the equation true.
The rule
The roots of a quadratic equation ax² + bx + c = 0 are given by the formula: x = (−b ± √(b² − 4ac)) / (2a)
Here, a = 1, b = 5, c = 6.
Let's understand it simply
Imagine you are rolling a roti (chapati) and want it to be perfectly round. If you press too hard or too soft, it goes out of shape. In maths, quadratic equations are like that—they have a balance point, called roots, where everything is just right (the equation equals zero). Our job is to find those points.
How it works
- Write the equation: x² + 5x + 6 = 0.
- Try to factor it: Find two numbers that multiply to 6 and add up to 5.
- The numbers are 2 and 3 (since 2 × 3 = 6 and 2 + 3 = 5).
- Rewrite the equation as (x + 2)(x + 3) = 0.
- Set each bracket to zero: x + 2 = 0 or x + 3 = 0.
- Solve for x: x = −2 or x = −3.
- Write the equation: x² + 5x + 6 = 0
- Find two numbers that multiply to 6 and add to 5
- Numbers are 2 and 3
- Rewrite as (x + 2)(x + 3) = 0
- Set each factor to zero: x + 2 = 0 or x + 3 = 0
- Solve: x = −2 or x = −3
Where you see it in real life
- Throwing a cricket ball: The path it follows is a quadratic curve. Solving the equation tells you where it will land.
- Profit in a shop: Sometimes, profit depends on price in a quadratic way. Roots show break-even points.
- Area of a rectangle: If you know the area and one side, you can use a quadratic equation to find the other side.
- Diwali rocket: The height at which it bursts can be found using a quadratic equation.
Why it matters
- Quadratic equations come up in science, engineering, and even in daily life calculations.
- Many board exam questions ask you to solve them.
- They help you understand more advanced maths, like calculus and physics.
- Knowing how to solve them saves time in competitive exams.
Worked example
Given: x² + 5x + 6 = 0
Step 1: Try to factor: Find two numbers that multiply to 6 and add to 5.
Step 2: 2 and 3 work (2 × 3 = 6, 2 + 3 = 5).
Step 3: Write as (x + 2)(x + 3) = 0
Step 4: Set each factor to zero: x + 2 = 0 or x + 3 = 0
Step 5: Solve: x = −2 or x = −3Common mistakes
- Forgetting to set the equation to zero before solving.
- Mixing up signs (writing x = 2 or x = 3 instead of x = −2 or x = −3).
- Not checking if factoring is possible; sometimes you need the quadratic formula.
Remember it
"Find two numbers that multiply to c and add to b."
Check yourself
What are the roots of x² + 5x + 6 = 0?
x = −2 and x = −3, because (−2)² + 5×(−2) + 6 = 0 and (−3)² + 5×(−3) + 6 = 0.
Why do we set the equation to zero before solving?
Because the roots are the values of x that make the equation equal to zero.
If you cannot factor a quadratic equation easily, what method should you use?
Use the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a).
Quick recap
- Quadratic equations can often be solved by factoring.
- Find two numbers that multiply to c and add to b.
- Always set the equation to zero before solving.
- Check your answer by plugging back into the equation.