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Solve 5x² + 4x - 9 = 0

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

x² +x += 0
5x2+4x−9=0⇒x=1, −955x^2 + 4x - 9 = 0 \Rightarrow x = 1,\ -\frac{9}{5}
Discriminant D = 196 · x ≈ 1, -1.8

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

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Graph of y = 5x² + 4x - 9-4-3-2-1123204060x = 1x = -1.8vertexy = 5x² + 4x - 9
Graph of y = 5x² + 4x - 9. The curve crosses the x-axis at x = 1 and x = -1.8; 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=42−4⋅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:

    (x−1)(5x+9)=0\left(x - 1\right)\left(5x + 9\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+9=0⇒x=−95x - 1 = 0 \Rightarrow x = 1\qquad 5x + 9 = 0 \Rightarrow x = -\frac{9}{5}
  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=1orx=−95x = 1\quad\text{or}\quad x = -\frac{9}{5}
  8. 8

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

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

The parabola y = 5x² + 4x - 9

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

Questions

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

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

Does 5x² + 4x - 9 = 0 factor?

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

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