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Solve 5x² + 6x - 8 = 0

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

x² +x += 0
5x2+6x−8=0⇒x=45, −25x^2 + 6x - 8 = 0 \Rightarrow x = \frac{4}{5},\ -2
Discriminant D = 196 · x ≈ 0.8, -2

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

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Graph of y = 5x² + 6x - 8-4-3-2-1123204060x = 0.8x = -2vertexy = 5x² + 6x - 8
Graph of y = 5x² + 6x - 8. The curve crosses the x-axis at x = 0.8 and x = -2; 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=62−4⋅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:

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

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

    5x−4=0⇒x=45x+2=0⇒x=−25x - 4 = 0 \Rightarrow x = \frac{4}{5}\qquad x + 2 = 0 \Rightarrow x = -2
  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=45orx=−2x = \frac{4}{5}\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = 4/5 back in: a·x² + b·x + c = 5·(16/25) + 6·(4/5) + (-8) = 0 ✓

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

The parabola y = 5x² + 6x - 8

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

Questions

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

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

Does 5x² + 6x - 8 = 0 factor?

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