What is a quadratic equation?
Before solving quadratic equation, it is necessary to understand the quadratic equation at first. A quadratic equation is an equation with an x² term. It looks like ax² + bx + c = 0. These equations pop up everywhere in maths. The quadratic formula is the easiest way to solve them every time.
Let’s solve 2x² + 5x – 3 = 0
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Step 1: Identify the coefficients
A quadratic equation has the standard form ax² + bx + c = 0
From our equation 2x² + 5x – 3 = 0:
- a = 2
- b = 5
- c = -3
Step 2: Write the quadratic formula
The quadratic formula is:
x = (-b ± √(b² – 4ac)) / (2a)
This formula gives you the solutions (roots) of any quadratic equation.
Step 3: Calculate the discriminant
The discriminant is the part under the square root: b² – 4ac
b² – 4ac = (5)² – 4(2)(-3) = 25 – (-24) = 25 + 24 = 49
The discriminant tells us important information:
- If it’s positive (like 49), we get two real solutions
- If it’s zero, we get one real solution
- If it’s negative, we get two complex solutions
Since 49 is positive, we have two real solutions.
Step 4: Calculate √(discriminant)
√49 = 7
Step 5: Apply the quadratic formula
x = (-b ± √(b² – 4ac)) / (2a)
x = (-5 ± 7) / (2 · 2)
x = (-5 ± 7) / 4
Step 6: Find both solutions
First solution (using +): x₁ = (-5 + 7) / 4 = 2/4 = 1/2 or 0.5
Second solution (using -): x₂ = (-5 – 7) / 4 = -12/4 = -3
Step 7: Verify the solutions
Let’s check x = 0.5: 2(0.5)² + 5(0.5) – 3 = 2(0.25) + 2.5 – 3 = 0.5 + 2.5 – 3 = 0 ✓
Let’s check x = -3: 2(-3)² + 5(-3) – 3 = 2(9) – 15 – 3 = 18 – 15 – 3 = 0 ✓
Final Answer: x = 0.5 or x = -3
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