WordGPT

Solve 5x² - 3x - 2 = 0

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

x² +x += 0
5x2−3x−2=0⇒x=1, −255x^2 - 3x - 2 = 0 \Rightarrow x = 1,\ -\frac{2}{5}
Discriminant D = 49 · x ≈ 1, -0.4

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

Explain it step by step in chatMake a short video of it
Graph of y = 5x² - 3x - 2-2-11231020304050x = 1x = -0.4vertexy = 5x² - 3x - 2
Graph of y = 5x² - 3x - 2. The curve crosses the x-axis at x = 1 and x = -0.4; the vertex is at (0.3, -2.45).

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 = -3, c = -2.

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

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

    D=(−3)2−4⋅5⋅(−2)=9+40=49D = (-3)^2 - 4\cdot5\cdot(-2) = 9 + 40 = 49
  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+2)=0\left(x - 1\right)\left(5x + 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:

    x−1=0⇒x=15x+2=0⇒x=−25x - 1 = 0 \Rightarrow x = 1\qquad 5x + 2 = 0 \Rightarrow x = -\frac{2}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±4910x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{49}}{10}
  7. 7

    Simplify to get the solutions:

    x=1orx=−25x = 1\quad\text{or}\quad x = -\frac{2}{5}
  8. 8

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

LaTeX source of the formulas on this page
\[ 5x^2 - 3x - 2 = 0\ \Rightarrow\ x = 1,\ -\frac{2}{5} \]
\[ 5x^2 - 3x - 2 = 0 \]
\[ D = (-3)^2 - 4\cdot5\cdot(-2) = 9 + 40 = 49 \]
\[ \left(x - 1\right)\left(5x + 2\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 5x + 2 = 0 \Rightarrow x = -\frac{2}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{49}}{10} \]
\[ x = 1\quad\text{or}\quad x = -\frac{2}{5} \]

The parabola y = 5x² - 3x - 2

xy
-2.742.55
-1.717.55
-0.72.55
0.3-2.45
1.32.55
2.317.55
3.342.55

Questions

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

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

Does 5x² - 3x - 2 = 0 factor?

Yes: (x-1)(5x+2) = 0.

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

(3/10, -49/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

Similar equations

    Solve 5x² - 3x - 2 = 0: x = 1 or x = -2/5 (steps) | WordGPT