WordGPT

Solve 5x² - 4x - 9 = 0

The solutions of 5x² - 4x - 9 = 0 are x = 9/5 or x = -1.

x² +x += 0
5x2−4x−9=0⇒x=95, −15x^2 - 4x - 9 = 0 \Rightarrow x = \frac{9}{5},\ -1
Discriminant D = 196 · x ≈ 1.8, -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 - 4x - 9-3-2-11234204060x = 1.8x = -1vertexy = 5x² - 4x - 9
Graph of y = 5x² - 4x - 9. The curve crosses the x-axis at x = 1.8 and x = -1; the vertex is at (0.4, -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 = -4, c = -9.

    5x2−4x−9=05x^2 - 4x - 9 = 0
  2. 2

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

    D=(−4)2−4⋅5⋅(−9)=16+180=196D = (-4)^2 - 4\cdot5\cdot(-9) = 16 + 180 = 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−9)(x+1)=0\left(5x - 9\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−9=0⇒x=95x+1=0⇒x=−15x - 9 = 0 \Rightarrow x = \frac{9}{5}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=4±19610x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{196}}{10}
  7. 7

    Simplify to get the solutions:

    x=95orx=−1x = \frac{9}{5}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 9/5 back in: a·x² + b·x + c = 5·(81/25) + (-4)·(9/5) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 4x - 9 = 0\ \Rightarrow\ x = \frac{9}{5},\ -1 \]
\[ 5x^2 - 4x - 9 = 0 \]
\[ D = (-4)^2 - 4\cdot5\cdot(-9) = 16 + 180 = 196 \]
\[ \left(5x - 9\right)\left(x + 1\right) = 0 \]
\[ 5x - 9 = 0 \Rightarrow x = \frac{9}{5}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{196}}{10} \]
\[ x = \frac{9}{5}\quad\text{or}\quad x = -1 \]

The parabola y = 5x² - 4x - 9

xy
-2.635.2
-1.610.2
-0.6-4.8
0.4-9.8
1.4-4.8
2.410.2
3.435.2

Questions

What are the solutions of 5x² - 4x - 9 = 0?

The solutions of 5x² - 4x - 9 = 0 are x = 9/5 or x = -1.

Does 5x² - 4x - 9 = 0 factor?

Yes: (5x-9)(x+1) = 0.

What is the vertex of y = 5x² - 4x - 9?

(2/5, -49/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 4/5, product = c/a = -9/5.

Similar equations

    Solve 5x² - 4x - 9 = 0: x = 9/5 or x = -1 (steps) | WordGPT