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Solve 5x² - 9x + 4 = 0

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

x² +x += 0
5x2−9x+4=0⇒x=1, 455x^2 - 9x + 4 = 0 \Rightarrow x = 1,\ \frac{4}{5}
Discriminant D = 1 · x ≈ 1, 0.8

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

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Graph of y = 5x² - 9x + 4-1.5-1-0.50.511.522.533.55101520253035x = 1x = 0.8vertexy = 5x² - 9x + 4
Graph of y = 5x² - 9x + 4. The curve crosses the x-axis at x = 1 and x = 0.8; the vertex is at (0.9, -0.05).

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 = -9, c = 4.

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

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

    D=(−9)2−4⋅5⋅4=81−80=1D = (-9)^2 - 4\cdot5\cdot4 = 81 - 80 = 1
  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−4)=0\left(x - 1\right)\left(5x - 4\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−4=0⇒x=45x - 1 = 0 \Rightarrow x = 1\qquad 5x - 4 = 0 \Rightarrow x = \frac{4}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=9±110x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{1}}{10}
  7. 7

    Simplify to get the solutions:

    x=1orx=45x = 1\quad\text{or}\quad x = \frac{4}{5}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 5·(1) + (-9)·(1) + 4 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 9x + 4 = 0\ \Rightarrow\ x = 1,\ \frac{4}{5} \]
\[ 5x^2 - 9x + 4 = 0 \]
\[ D = (-9)^2 - 4\cdot5\cdot4 = 81 - 80 = 1 \]
\[ \left(x - 1\right)\left(5x - 4\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 5x - 4 = 0 \Rightarrow x = \frac{4}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{1}}{10} \]
\[ x = 1\quad\text{or}\quad x = \frac{4}{5} \]

The parabola y = 5x² - 9x + 4

xy
-2.144.95
-1.119.95
-0.14.95
0.9-0.05
1.94.95
2.919.95
3.944.95

Questions

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

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

Does 5x² - 9x + 4 = 0 factor?

Yes: (x-1)(5x-4) = 0.

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

(9/10, -1/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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