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Discriminant with a leading coefficient, step by step

When the leading coefficient is not 1, b² − 4ac still decides it — but the a can no longer be ignored.

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The question

How many real solutions?

8x² − 16x − 90 = 0

8x² − 16x − 90 = 0
a = 8b = −16c = −90
b² − 4ac = (−16)² − 4(8)(−90)
= 3136 > 0→two
  1. You do not have to solve this to answer it. One number decides how many real solutions a quadratic has, and it is b² − 4ac — the part that ends up under the square root sign when you do solve it.
  2. Read off a, b and c from 8x² − 16x − 90 = 0: a = 8, b = −16, c = −90. Write all three in, even the a — on this sheet it is 8, and forgetting it is the mistake that only shows up when it is not 1.
  3. Work it out: (−16)² is 256, and 4 × 8 × −90 is −2880. That leaves 256 − −2880 = 3136.
  4. 3136 is above zero, and a positive number has two square roots — one either way from the ±. So there are TWO real solutions.

How it works.

  1. 01

    You do not have to solve this to answer it. One number decides how many real solutions a quadratic has, and it is b² − 4ac — the part that ends up under the square root sign when you do solve it.

  2. 02

    Read off a, b and c from 8x² − 16x − 90 = 0: a = 8, b = −16, c = −90. Write all three in, even the a — on this sheet it is 8, and forgetting it is the mistake that only shows up when it is not 1.

  3. 03

    Work it out: (−16)² is 256, and 4 × 8 × −90 is −2880. That leaves 256 − −2880 = 3136.

  4. 04

    3136 is above zero, and a positive number has two square roots — one either way from the ±. So there are TWO real solutions.

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