WordGPT

Solve 5x² + 3x - 2 = 0

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

x² +x += 0
5x2+3x−2=0⇒x=25, −15x^2 + 3x - 2 = 0 \Rightarrow x = \frac{2}{5},\ -1
Discriminant D = 49 · x ≈ 0.4, -1

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 - 2-3-2-1121020304050x = 0.4x = -1vertexy = 5x² + 3x - 2
Graph of y = 5x² + 3x - 2. The curve crosses the x-axis at x = 0.4 and x = -1; 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=32−4⋅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:

    (5x−2)(x+1)=0\left(5x - 2\right)\left(x + 1\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−2=0⇒x=25x+1=0⇒x=−15x - 2 = 0 \Rightarrow x = \frac{2}{5}\qquad x + 1 = 0 \Rightarrow x = -1
  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=25orx=−1x = \frac{2}{5}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 2/5 back in: a·x² + b·x + c = 5·(4/25) + 3·(2/5) + (-2) = 0 ✓

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

The parabola y = 5x² + 3x - 2

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

Questions

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

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

Does 5x² + 3x - 2 = 0 factor?

Yes: (5x-2)(x+1) = 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 = 2/5 or x = -1 (steps) | WordGPT