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Solve 4x² - 5x - 9 = 0

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

x² +x += 0
4x2−5x−9=0⇒x=94, −14x^2 - 5x - 9 = 0 \Rightarrow x = \frac{9}{4},\ -1
Discriminant D = 169 · x ≈ 2.25, -1

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

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Graph of y = 4x² - 5x - 9-3-2-11234-10102030405060x = 2.25x = -1vertexy = 4x² - 5x - 9
Graph of y = 4x² - 5x - 9. The curve crosses the x-axis at x = 2.25 and x = -1; the vertex is at (0.625, -10.5625).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = -5, c = -9.

    4x2−5x−9=04x^2 - 5x - 9 = 0
  2. 2

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

    D=(−5)2−4⋅4⋅(−9)=25+144=169D = (-5)^2 - 4\cdot4\cdot(-9) = 25 + 144 = 169
  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:

    (4x−9)(x+1)=0\left(4x - 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:

    4x−9=0⇒x=94x+1=0⇒x=−14x - 9 = 0 \Rightarrow x = \frac{9}{4}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±1698x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{169}}{8}
  7. 7

    Simplify to get the solutions:

    x=94orx=−1x = \frac{9}{4}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 9/4 back in: a·x² + b·x + c = 4·(81/16) + (-5)·(9/4) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 5x - 9 = 0\ \Rightarrow\ x = \frac{9}{4},\ -1 \]
\[ 4x^2 - 5x - 9 = 0 \]
\[ D = (-5)^2 - 4\cdot4\cdot(-9) = 25 + 144 = 169 \]
\[ \left(4x - 9\right)\left(x + 1\right) = 0 \]
\[ 4x - 9 = 0 \Rightarrow x = \frac{9}{4}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{169}}{8} \]
\[ x = \frac{9}{4}\quad\text{or}\quad x = -1 \]

The parabola y = 4x² - 5x - 9

xy
-2.37525.4375
-1.3755.4375
-0.375-6.5625
0.625-10.5625
1.625-6.5625
2.6255.4375
3.62525.4375

Questions

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

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

Does 4x² - 5x - 9 = 0 factor?

Yes: (4x-9)(x+1) = 0.

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

(5/8, -169/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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