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Solve 5x² - 6x - 8 = 0

The solutions of 5x² - 6x - 8 = 0 are x = 2 or x = -4/5.

x² +x += 0
5x2−6x−8=0⇒x=2, −455x^2 - 6x - 8 = 0 \Rightarrow x = 2,\ -\frac{4}{5}
Discriminant D = 196 · x ≈ 2, -0.8

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = 5x² - 6x - 8-3-2-11234204060x = 2x = -0.8vertexy = 5x² - 6x - 8
Graph of y = 5x² - 6x - 8. The curve crosses the x-axis at x = 2 and x = -0.8; the vertex is at (0.6, -9.8).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = -6, c = -8.

    5x2−6x−8=05x^2 - 6x - 8 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=(−6)2−4⋅5⋅(−8)=36+160=196D = (-6)^2 - 4\cdot5\cdot(-8) = 36 + 160 = 196
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (x−2)(5x+4)=0\left(x - 2\right)\left(5x + 4\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x−2=0⇒x=25x+4=0⇒x=−45x - 2 = 0 \Rightarrow x = 2\qquad 5x + 4 = 0 \Rightarrow x = -\frac{4}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=6±19610x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{196}}{10}
  7. 7

    Simplify to get the solutions:

    x=2orx=−45x = 2\quad\text{or}\quad x = -\frac{4}{5}
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = 5·(4) + (-6)·(2) + (-8) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 6x - 8 = 0\ \Rightarrow\ x = 2,\ -\frac{4}{5} \]
\[ 5x^2 - 6x - 8 = 0 \]
\[ D = (-6)^2 - 4\cdot5\cdot(-8) = 36 + 160 = 196 \]
\[ \left(x - 2\right)\left(5x + 4\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 5x + 4 = 0 \Rightarrow x = -\frac{4}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{196}}{10} \]
\[ x = 2\quad\text{or}\quad x = -\frac{4}{5} \]

The parabola y = 5x² - 6x - 8

xy
-2.435.2
-1.410.2
-0.4-4.8
0.6-9.8
1.6-4.8
2.610.2
3.635.2

Questions

What are the solutions of 5x² - 6x - 8 = 0?

The solutions of 5x² - 6x - 8 = 0 are x = 2 or x = -4/5.

Does 5x² - 6x - 8 = 0 factor?

Yes: (x-2)(5x+4) = 0.

What is the vertex of y = 5x² - 6x - 8?

(3/5, -49/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6/5, product = c/a = -8/5.

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