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Using the quadratic formula and the discriminant
Learn step-by-step with practice exercises built right in.
The quadratic formula is a powerful tool that solves ANY quadratic equation, even when factoring is difficult or impossible.
For any quadratic equation in standard form: axยฒ + bx + c = 0
The solutions are given by:
x = (-b ยฑ โ(bยฒ - 4ac)) / (2a)
This formula ALWAYS works for quadratic equations!
Let's break down each part:
-b: The opposite of the coefficient of x
ยฑ: Plus-or-minus symbol (gives us two solutions)
โ(bยฒ - 4ac): The square root of the discriminant
2a: Twice the coefficient of xยฒ
The discriminant: bยฒ - 4ac This tells us about the nature of the solutions.
Use the quadratic formula when:
Use the discriminant to determine the number of real solutions:
Avoid these 3 frequent errors
See how this math is used in the real world
Solve .
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You might use other methods when:
Step 1: Write equation in standard form (axยฒ + bx + c = 0)
Step 2: Identify a, b, and c
Step 3: Substitute into the formula
Step 4: Simplify under the square root (the discriminant)
Step 5: Simplify the entire expression
Step 6: Write two solutions (one with +, one with -)
Step 7: Check your solutions
Solve: xยฒ + 5x + 3 = 0
Step 1: Already in standard form
Step 2: Identify coefficients
Step 3: Substitute into formula x = (-5 ยฑ โ(5ยฒ - 4(1)(3))) / (2(1))
Step 4: Simplify discriminant x = (-5 ยฑ โ(25 - 12)) / 2 x = (-5 ยฑ โ13) / 2
Step 5: This is simplified (โ13 cannot be simplified)
Step 6: Two solutions x = (-5 + โ13) / 2 or x = (-5 - โ13) / 2
Approximate values: x โ (-5 + 3.606) / 2 โ -0.697 x โ (-5 - 3.606) / 2 โ -4.303
Solve: xยฒ - 4x + 1 = 0
Coefficients: a = 1, b = -4, c = 1
Substitute: x = (4 ยฑ โ((-4)ยฒ - 4(1)(1))) / (2(1)) x = (4 ยฑ โ(16 - 4)) / 2 x = (4 ยฑ โ12) / 2
Simplify โ12: โ12 = โ(4 ยท 3) = 2โ3
x = (4 ยฑ 2โ3) / 2
Factor out 2: x = 2(2 ยฑ โ3) / 2 x = 2 ยฑ โ3
Solutions: x = 2 + โ3 or x = 2 - โ3
Solve: 2xยฒ + 7x + 3 = 0
Coefficients: a = 2, b = 7, c = 3
Substitute: x = (-7 ยฑ โ(7ยฒ - 4(2)(3))) / (2(2)) x = (-7 ยฑ โ(49 - 24)) / 4 x = (-7 ยฑ โ25) / 4 x = (-7 ยฑ 5) / 4
Two solutions: x = (-7 + 5) / 4 = -2/4 = -1/2 x = (-7 - 5) / 4 = -12/4 = -3
Check: This could have been factored as (2x + 1)(x + 3) = 0
Solve: 3xยฒ = 5x + 2
Step 1: Standard form 3xยฒ - 5x - 2 = 0
Coefficients: a = 3, b = -5, c = -2
Substitute: x = (5 ยฑ โ((-5)ยฒ - 4(3)(-2))) / (2(3)) x = (5 ยฑ โ(25 + 24)) / 6 x = (5 ยฑ โ49) / 6 x = (5 ยฑ 7) / 6
Solutions: x = (5 + 7) / 6 = 12/6 = 2 x = (5 - 7) / 6 = -2/6 = -1/3
The discriminant tells us the nature of solutions BEFORE we solve!
If bยฒ - 4ac > 0: Two different real solutions
If bยฒ - 4ac = 0: One repeated real solution (two equal solutions)
If bยฒ - 4ac < 0: No real solutions (two complex solutions)
Example 1: How many solutions does xยฒ + 6x + 5 = 0 have?
Calculate discriminant: bยฒ - 4ac = 6ยฒ - 4(1)(5) = 36 - 20 = 16
Since 16 > 0, there are two different real solutions.
We could solve: x = (-6 ยฑ 4) / 2, giving x = -1 or x = -5
Example 2: How many solutions does xยฒ - 4x + 4 = 0 have?
bยฒ - 4ac = (-4)ยฒ - 4(1)(4) = 16 - 16 = 0
Since discriminant = 0, there is one repeated solution.
x = (4 ยฑ 0) / 2 = 2
This is (x - 2)ยฒ = 0
Example 3: How many solutions does xยฒ + 2x + 5 = 0 have?
bยฒ - 4ac = 2ยฒ - 4(1)(5) = 4 - 20 = -16
Since -16 < 0, there are no real solutions.
The graph doesn't cross the x-axis.
When the discriminant equals zero, you get one repeated root.
Example: xยฒ - 10x + 25 = 0
a = 1, b = -10, c = 25
Discriminant: (-10)ยฒ - 4(1)(25) = 100 - 100 = 0
x = (10 ยฑ 0) / 2 = 10/2 = 5
This is a perfect square: (x - 5)ยฒ = 0
Be extra careful when b or c is negative!
Example: xยฒ - 3x - 4 = 0
a = 1, b = -3, c = -4
x = (-(-3) ยฑ โ((-3)ยฒ - 4(1)(-4))) / (2(1)) x = (3 ยฑ โ(9 + 16)) / 2 x = (3 ยฑ โ25) / 2 x = (3 ยฑ 5) / 2
x = 4 or x = -1
Common mistakes:
Always simplify radicals in your answer.
Example: If you get โ50: โ50 = โ(25 ยท 2) = 5โ2
Example: If you get โ72: โ72 = โ(36 ยท 2) = 6โ2
Example: x = (6 ยฑ โ48) / 4
Simplify โ48 = โ(16 ยท 3) = 4โ3
x = (6 ยฑ 4โ3) / 4
Factor out 2: x = 2(3 ยฑ 2โ3) / 4 x = (3 ยฑ 2โ3) / 2
The height of a projectile is often modeled by: h = -16tยฒ + vโt + hโ
Example: A ball is thrown upward at 64 ft/s from height 6 ft. When does it hit the ground?
Equation: -16tยฒ + 64t + 6 = 0
Divide by -2: 8tยฒ - 32t - 3 = 0
Using quadratic formula: a = 8, b = -32, c = -3
t = (32 ยฑ โ(1024 + 96)) / 16 t = (32 ยฑ โ1120) / 16 t = (32 ยฑ 4โ70) / 16 t = (8 ยฑ โ70) / 4
t โ (8 + 8.37) / 4 โ 4.09 seconds (positive solution)
We reject the negative time solution.
Example: A rectangle has length 4 cm more than its width. Area is 60 cmยฒ. Find dimensions.
Let w = width Then w + 4 = length
w(w + 4) = 60 wยฒ + 4w - 60 = 0
Using quadratic formula: w = (-4 ยฑ โ(16 + 240)) / 2 w = (-4 ยฑ โ256) / 2 w = (-4 ยฑ 16) / 2
w = 6 or w = -10
Since width must be positive: w = 6 cm, length = 10 cm
Example: The sum of a number and its reciprocal is 13/6. Find the number.
Let x = the number Then 1/x = reciprocal
x + 1/x = 13/6
Multiply by 6x: 6xยฒ + 6 = 13x Rearrange: 6xยฒ - 13x + 6 = 0
x = (13 ยฑ โ(169 - 144)) / 12 x = (13 ยฑ โ25) / 12 x = (13 ยฑ 5) / 12
x = 18/12 = 3/2 or x = 8/12 = 2/3
Both solutions work! (They're reciprocals of each other)
Substitute back into the original equation to verify.
Example: For xยฒ - 5x + 6 = 0, we get x = 2 or x = 3
Check x = 2: 2ยฒ - 5(2) + 6 = 4 - 10 + 6 = 0 โ Check x = 3: 3ยฒ - 5(3) + 6 = 9 - 15 + 6 = 0 โ
Forgetting the negative sign in -b The formula is -b, not b!
Order of operations in discriminant Calculate bยฒ first, then 4ac, then subtract
Not simplifying radicals โ12 should be written as 2โ3
Forgetting ยฑ gives TWO solutions Don't just use + or just use -
Division errors Divide ENTIRE numerator by denominator
Sign errors with negative b or c Be extra careful: -(-3) = 3, -4(-2) = 8
Factoring:
Quadratic Formula:
Completing the Square:
Level 1: a = 1, perfect squares
Level 2: a = 1, two solutions
Level 3: a โ 1
Level 4: Requires simplifying
Level 5: Applications
The Formula: x = (-b ยฑ โ(bยฒ - 4ac)) / (2a)
The Discriminant: bยฒ - 4ac
Steps:
Identify: , ,
Calculate the discriminant:
Since , there is one real solution (a repeated root).
Answer: One real solution
Solve using the quadratic formula:
Identify: , ,
Substitute into the quadratic formula:
Solve:
Identify: , ,
Substitute:
Two solutions:
Answer: or
This cannot be simplified further.
Answer: or
(Approximately: or )