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Algebra II

The Quadratic Formula

Lesson

The quadratic formulasolves any quadratic equation — even ones that don’t factor nicely. Given:

ax2+bx+c=0ax^2 + bx + c = 0

the solutions are:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The ±\pm means there are two solutions: one with + on top of the radical, one with −. The expression under the radical (b2−4ac)(b^2 - 4ac) is called the discriminant.

To use the formula:

  1. Make sure the equation is in the form ax2+bx+c=0ax^2 + bx + c = 0 (everything on one side, zero on the other).
  2. Identify aa, bb, and cc, including signs.
  3. Plug into the formula and simplify.

Worked example 1

x2+5x+6=0x^2 + 5x + 6 = 0

Identify: a=1,b=5,c=6a = 1, b = 5, c = 6.

x=−5±25−242=−5±12x = \frac{-5 \pm \sqrt{25 - 24}}{2} = \frac{-5 \pm 1}{2}

Two solutions:

x=−5+12=−2orx=−5−12=−3x = \frac{-5 + 1}{2} = -2 \quad \text{or} \quad x = \frac{-5 - 1}{2} = -3

Worked example 2

2x2−7x+3=02x^2 - 7x + 3 = 0

a=2,b=−7,c=3a = 2, b = -7, c = 3.

x=7±49−244=7±54x = \frac{7 \pm \sqrt{49 - 24}}{4} = \frac{7 \pm 5}{4}
x=3orx=12x = 3 \quad \text{or} \quad x = \frac{1}{2}

How to type your answer

Enter both solutions separated by a comma— order doesn’t matter. Use fractions where helpful (e.g. 1/2). Examples: -2,-3, 3,1/2, 5,-1.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

x2+5x+6=0x^2 + 5x + 6 = 0

Problem 2

x2−5x+6=0x^2 - 5x + 6 = 0

Problem 3

x2+x−12=0x^2 + x - 12 = 0

Problem 4

x2−9=0x^2 - 9 = 0

Practice

Standard problems matching the lesson.

Problem 5

x2+7x+10=0x^2 + 7x + 10 = 0

Problem 6

x2−8x+15=0x^2 - 8x + 15 = 0

Problem 7

x2−4x−21=0x^2 - 4x - 21 = 0

Problem 8

x2+6x+5=0x^2 + 6x + 5 = 0

Problem 9

2x2−7x+3=02x^2 - 7x + 3 = 0

Problem 10

2x2+5x−3=02x^2 + 5x - 3 = 0

Problem 11

3x2−7x+2=03x^2 - 7x + 2 = 0

Problem 12

x2+2x−15=0x^2 + 2x - 15 = 0

Problem 13

x2−10x+25=0x^2 - 10x + 25 = 0

Problem 14

x2+4x+4=0x^2 + 4x + 4 = 0

Problem 15

2x2−5x−3=02x^2 - 5x - 3 = 0

Problem 16

3x2+4x+1=03x^2 + 4x + 1 = 0

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

x2=5x+14x^2 = 5x + 14

Problem 18

x2+9x=−20x^2 + 9x = -20

Problem 19

4x2−4x+1=04x^2 - 4x + 1 = 0

Problem 20

6x2+x−1=06x^2 + x - 1 = 0

Problem 21

x2−6x+8=0x^2 - 6x + 8 = 0

Problem 22

5x2−14x−3=05x^2 - 14x - 3 = 0

Practice

Standard problems matching the lesson.

Problem 23

A ball thrown straight up: h(t) = -5t² + 30t (meters). Solve h(t) = 0 for both values of t.

Problem 24

A rectangle has length x and width (x − 1) with area 12. Solve x(x − 1) = 12 for both values of x.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Two consecutive integers n and (n + 1) multiply to 56. Solve n(n + 1) = 56 for both n.

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