The quadratic formula is a reliable, universal tool for solving any quadratic equation of the form:
ax² + bx + c = 0
No matter how tricky the coefficients, this formula gives you the exact roots every time:
x = (-b ± √(b² – 4ac)) / 2a
In this post, we’ll walk through several examples—from simple to more complex—to show you how to solve quadratic equations step-by-step using the quadratic formula.
🔢 Example 1: Simple Quadratic with Two Real Roots
Solve:
x² – 3x – 4 = 0
Step 1: Identify coefficients
a = 1, b = -3, c = -4
Step 2: Calculate the discriminant
Δ = b² – 4ac = (-3)² – 4(1)(-4) = 9 + 16 = 25
Step 3: Apply the quadratic formula
x = [-(-3) ± √25] / (2 × 1)
x = [3 ± 5] / 2
Step 4: Find both roots
x₁ = (3 + 5)/2 = 8/2 = 4
x₂ = (3 – 5)/2 = (-2)/2 = -1
Final answer:
x = 4 or x = -1
🔢 Example 2: Quadratic with One Real Root (Repeated Root)
Solve:
4x² – 4x + 1 = 0
Step 1: Identify coefficients
a = 4, b = -4, c = 1
Step 2: Calculate the discriminant
Δ = (-4)² – 4(4)(1) = 16 – 16 = 0
Step 3: Apply the quadratic formula
x = [-(-4) ± √0] / (2 × 4)
x = (4 ± 0) / 8 = 4/8 = 0.5
Final answer:
x = 0.5 (a repeated root)
🔢 Example 3: Quadratic with Complex Roots
Solve:
x² + 2x + 5 = 0
Step 1: Identify coefficients
a = 1, b = 2, c = 5
Step 2: Calculate the discriminant
Δ = 2² – 4(1)(5) = 4 – 20 = -16 (negative discriminant means complex roots)
Step 3: Apply the quadratic formula
x = [-2 ± √(-16)] / 2
x = [-2 ± 4i] / 2 (since √-16 = 4i, where i is the imaginary unit)
Step 4: Simplify
x₁ = (-2 + 4i)/2 = -1 + 2i
x₂ = (-2 – 4i)/2 = -1 – 2i
Final answer:
x = -1 + 2i or x = -1 – 2i
🔢 Example 4: Quadratic with Fractional Coefficients
Solve:
(1/2)x² – (3/4)x + (1/8) = 0
Step 1: Identify coefficients
a = 1/2, b = -3/4, c = 1/8
Step 2: Calculate the discriminant
Δ = (-3/4)² – 4(1/2)(1/8)
= 9/16 – 4 × 1/2 × 1/8
= 9/16 – 1/4
= 9/16 – 4/16 = 5/16
Step 3: Apply the quadratic formula
x = [3/4 ± √(5/16)] / (2 × 1/2)
x = [3/4 ± (√5)/4] / 1
Step 4: Simplify roots
x₁ = (3/4 + √5/4) = (3 + √5)/4
x₂ = (3/4 – √5/4) = (3 – √5)/4
Final answer:
x = (3 + √5)/4 or x = (3 – √5)/4
🔢 Tips for Solving Quadratics Using the Formula
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Always identify a, b, and c correctly from the equation.
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Carefully compute the discriminant; it tells you the number and type of roots.
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Use parentheses when plugging values into the formula to avoid sign errors.
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Simplify square roots where possible.
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Remember to divide the entire numerator by 2a.
🎯 Conclusion
The quadratic formula is a dependable, go-to method for solving any quadratic equation—simple or complex. By practicing with varied examples, you’ll become comfortable using it quickly and accurately.
No matter the values of a, b, or c, the quadratic formula will always guide you to the roots of the equation.
