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Solve -2x² + 9x - 7 = 0

The solutions of -2x² + 9x - 7 = 0 are x = 1 or x = 7/2.

x² +x += 0
−2x2+9x−7=0⇒x=1, 72-2x^2 + 9x - 7 = 0 \Rightarrow x = 1,\ \frac{7}{2}
Discriminant D = 25 · x ≈ 1, 3.5

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

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Graph of y = -2x² + 9x - 7-1123456-25-20-15-10-55x = 1x = 3.5vertexy = -2x² + 9x - 7
Graph of y = -2x² + 9x - 7. The curve crosses the x-axis at x = 1 and x = 3.5; the vertex is at (2.25, 3.125).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = 9, c = -7.

    −2x2+9x−7=0-2x^2 + 9x - 7 = 0
  2. 2

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

    D=92−4⋅(−2)⋅(−7)=81−56=25D = 9^2 - 4\cdot(-2)\cdot(-7) = 81 - 56 = 25
  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)(2x−7)=0-\left(x - 1\right)\left(2x - 7\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=12x−7=0⇒x=72x - 1 = 0 \Rightarrow x = 1\qquad 2x - 7 = 0 \Rightarrow x = \frac{7}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−9±25−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{25}}{-4}
  7. 7

    Simplify to get the solutions:

    x=1orx=72x = 1\quad\text{or}\quad x = \frac{7}{2}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = -2·(1) + 9·(1) + (-7) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 9x - 7 = 0\ \Rightarrow\ x = 1,\ \frac{7}{2} \]
\[ -2x^2 + 9x - 7 = 0 \]
\[ D = 9^2 - 4\cdot(-2)\cdot(-7) = 81 - 56 = 25 \]
\[ -\left(x - 1\right)\left(2x - 7\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 2x - 7 = 0 \Rightarrow x = \frac{7}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{25}}{-4} \]
\[ x = 1\quad\text{or}\quad x = \frac{7}{2} \]

The parabola y = -2x² + 9x - 7

xy
-0.75-14.875
0.25-4.875
1.251.125
2.253.125
3.251.125
4.25-4.875
5.25-14.875

Questions

What are the solutions of -2x² + 9x - 7 = 0?

The solutions of -2x² + 9x - 7 = 0 are x = 1 or x = 7/2.

Does -2x² + 9x - 7 = 0 factor?

Yes: -1(x-1)(2x-7) = 0.

What is the vertex of y = -2x² + 9x - 7?

(9/4, 25/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 9/2, product = c/a = 7/2.

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