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Solve 5x² - 2x - 3 = 0

The solutions of 5x² - 2x - 3 = 0 are x = 1 or x = -3/5.

x² +x += 0
5x2−2x−3=0⇒x=1, −355x^2 - 2x - 3 = 0 \Rightarrow x = 1,\ -\frac{3}{5}
Discriminant D = 64 · x ≈ 1, -0.6

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

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Graph of y = 5x² - 2x - 3-3-2-1123-101020304050x = 1x = -0.6vertexy = 5x² - 2x - 3
Graph of y = 5x² - 2x - 3. The curve crosses the x-axis at x = 1 and x = -0.6; the vertex is at (0.2, -3.2).

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 = -2, c = -3.

    5x2−2x−3=05x^2 - 2x - 3 = 0
  2. 2

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

    D=(−2)2−4⋅5⋅(−3)=4+60=64D = (-2)^2 - 4\cdot5\cdot(-3) = 4 + 60 = 64
  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−1)(5x+3)=0\left(x - 1\right)\left(5x + 3\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−1=0⇒x=15x+3=0⇒x=−35x - 1 = 0 \Rightarrow x = 1\qquad 5x + 3 = 0 \Rightarrow x = -\frac{3}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=2±6410x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{64}}{10}
  7. 7

    Simplify to get the solutions:

    x=1orx=−35x = 1\quad\text{or}\quad x = -\frac{3}{5}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 5·(1) + (-2)·(1) + (-3) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 2x - 3 = 0\ \Rightarrow\ x = 1,\ -\frac{3}{5} \]
\[ 5x^2 - 2x - 3 = 0 \]
\[ D = (-2)^2 - 4\cdot5\cdot(-3) = 4 + 60 = 64 \]
\[ \left(x - 1\right)\left(5x + 3\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 5x + 3 = 0 \Rightarrow x = -\frac{3}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{64}}{10} \]
\[ x = 1\quad\text{or}\quad x = -\frac{3}{5} \]

The parabola y = 5x² - 2x - 3

xy
-2.841.8
-1.816.8
-0.81.8
0.2-3.2
1.21.8
2.216.8
3.241.8

Questions

What are the solutions of 5x² - 2x - 3 = 0?

The solutions of 5x² - 2x - 3 = 0 are x = 1 or x = -3/5.

Does 5x² - 2x - 3 = 0 factor?

Yes: (x-1)(5x+3) = 0.

What is the vertex of y = 5x² - 2x - 3?

(1/5, -16/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 2/5, product = c/a = -3/5.

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