Solve -2x² + 11x - 9 = 0

The solutions of -2x² + 11x - 9 = 0 are x = 1 or x = 9/2.

x² +x += 0

-2x^2 + 11x - 9 = 0 \Rightarrow x = 1,\ \frac{9}{2}

Discriminant D = 49 · x ≈ 1, 4.5

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

-2x^2 + 11x - 9 = 0\ \Rightarrow\ x = 1,\ \frac{9}{2}

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Graph of y = -2x² + 11x - 9. The curve crosses the x-axis at x = 1 and x = 4.5; the vertex is at (2.75, 6.125).

How to work it out

1
Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = 11, c = -9.
$-2x^2 + 11x - 9 = 0$
2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
$D = 11^2 - 4\cdot(-2)\cdot(-9) = 121 - 72 = 49$
3
D is positive, so there are two different real solutions.
4
Method 1, factoring. The roots are rational, so the left side factors:
$-\left(x - 1\right)\left(2x - 9\right) = 0$
5
A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:
$x - 1 = 0 \Rightarrow x = 1\qquad 2x - 9 = 0 \Rightarrow x = \frac{9}{2}$
6
Method 2, The quadratic formula works for every quadratic:
$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{49}}{-4}$
7
Simplify to get the solutions:
$x = 1\quad\text{or}\quad x = \frac{9}{2}$
8
Check by putting x = 1 back in: a·x² + b·x + c = -2·(1) + 11·(1) + (-9) = 0 ✓

LaTeX source of the formulas on this page

\[ -2x^2 + 11x - 9 = 0\ \Rightarrow\ x = 1,\ \frac{9}{2} \]
\[ -2x^2 + 11x - 9 = 0 \]
\[ D = 11^2 - 4\cdot(-2)\cdot(-9) = 121 - 72 = 49 \]
\[ -\left(x - 1\right)\left(2x - 9\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 2x - 9 = 0 \Rightarrow x = \frac{9}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{49}}{-4} \]
\[ x = 1\quad\text{or}\quad x = \frac{9}{2} \]

The parabola y = -2x² + 11x - 9

x	y
-0.25	-11.875
0.75	-1.875
1.75	4.125
2.75	6.125
3.75	4.125
4.75	-1.875
5.75	-11.875

Questions

What are the solutions of -2x² + 11x - 9 = 0?

The solutions of -2x² + 11x - 9 = 0 are x = 1 or x = 9/2.

Does -2x² + 11x - 9 = 0 factor?

Yes: -1(x-1)(2x-9) = 0.

What is the vertex of y = -2x² + 11x - 9?

(11/4, 49/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 11/2, product = c/a = 9/2.

Solve -2x² + 11x - 9 = 0

How to work it out

The parabola y = -2x² + 11x - 9

Questions

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