The quadratic formula and the discriminant

About this lesson

Algebra I › Quadratic equations

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The quadratic formula and the discriminant

Warm-up

Three quick questions from earlier lessons.

Discover

x2+2x+5=0x^2 + 2x + 5 = 0. Completing the square gives (x+1)2=−4(x + 1)^2 = -4. How many real solutions?

  1. None
  2. Two

Key idea

The quadratic formula solves every quadratic equation.

The solutions of ax2+bx+c=0ax^2 + bx + c = 0 are x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}. The discriminant b2−4acb^2 - 4ac tells how many real solutions: positive, two; zero, one; negative, none.

Worked example

Solve 2x2+3x−2=02x^2 + 3x - 2 = 0.

b2−4ac=9+16=25b^2 - 4ac = 9 + 16 = 25
a = 2, b = 3, c = −2.
x=−3±54x = \dfrac{-3 \pm 5}{4}
Formula.
x=12x = \tfrac{1}{2} or x=−2x = -2
Two solutions.

Your turn, guided

Your turn: the discriminant of x2+5x+6=0x^2 + 5x + 6 = 0.

11
25 − 24.

Practice ★

  1. Find the discriminant b2−4acb^2 - 4ac of 1x2+12x+2=01x^2 + 12x + 2 = 0.
  2. Find the discriminant b2−4acb^2 - 4ac of 1x2+11x+9=01x^2 + 11x + 9 = 0.
  1. How many real solutions has x2+7x+49=0x^2 + 7x + 49 = 0?
    1. None
    2. Two
    3. One

Watch out!

Practice ★★

  1. Solve 2x2+5x−1=02x^2 + 5x - 1 = 0 with the quadratic formula. Enter all solutions (use sqrt).
  2. Solve 2x2+7x−6=02x^2 + 7x - 6 = 0 with the quadratic formula. Enter all solutions (use sqrt).
  1. Boss problem. A rectangular garden is 7 m longer than it is wide, and its area is 294 m². Let ww be the width in meters.

Exit ticket

Three questions to check you have got it.

  1. Find the discriminant b2−4acb^2 - 4ac of 1x2+12x+2=01x^2 + 12x + 2 = 0.
  2. Find the discriminant b2−4acb^2 - 4ac of 1x2+11x+9=01x^2 + 11x + 9 = 0.
  3. Find the discriminant b2−4acb^2 - 4ac of 2x2+8x+5=02x^2 + 8x + 5 = 0.

Summary card

The quadratic formula

x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}; the discriminant counts the solutions.

Watch out: 2a2a divides the whole numerator.

End of the lesson

Next lesson: Modeling with quadratic equations Back to the chapter

Answers: The quadratic formula and the discriminant

A

  1. 136136
  2. 8585

B

  1. None

C

  1. −54+334, −334−54- \frac{5}{4} + \frac{\sqrt{33}}{4},\ - \frac{\sqrt{33}}{4} - \frac{5}{4}
  2. −74+974, −974−74- \frac{7}{4} + \frac{\sqrt{97}}{4},\ - \frac{\sqrt{97}}{4} - \frac{7}{4}

D

  1. 2121

E

  1. 136136
  2. 8585
  3. 2424

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