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Solve -2x² + 7x - 5 = 0

The solutions of -2x² + 7x - 5 = 0 are x = 1 or x = 5/2.

x² +x += 0
−2x2+7x−5=0⇒x=1, 52-2x^2 + 7x - 5 = 0 \Rightarrow x = 1,\ \frac{5}{2}
Discriminant D = 9 · x ≈ 1, 2.5

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

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Graph of y = -2x² + 7x - 5-112345-20-15-10-5x = 1x = 2.5vertexy = -2x² + 7x - 5
Graph of y = -2x² + 7x - 5. The curve crosses the x-axis at x = 1 and x = 2.5; the vertex is at (1.75, 1.125).

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 = 7, c = -5.

    −2x2+7x−5=0-2x^2 + 7x - 5 = 0
  2. 2

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

    D=72−4⋅(−2)⋅(−5)=49−40=9D = 7^2 - 4\cdot(-2)\cdot(-5) = 49 - 40 = 9
  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−5)=0-\left(x - 1\right)\left(2x - 5\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−5=0⇒x=52x - 1 = 0 \Rightarrow x = 1\qquad 2x - 5 = 0 \Rightarrow x = \frac{5}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

    x=1orx=52x = 1\quad\text{or}\quad x = \frac{5}{2}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = -2·(1) + 7·(1) + (-5) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 7x - 5 = 0\ \Rightarrow\ x = 1,\ \frac{5}{2} \]
\[ -2x^2 + 7x - 5 = 0 \]
\[ D = 7^2 - 4\cdot(-2)\cdot(-5) = 49 - 40 = 9 \]
\[ -\left(x - 1\right)\left(2x - 5\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 2x - 5 = 0 \Rightarrow x = \frac{5}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{9}}{-4} \]
\[ x = 1\quad\text{or}\quad x = \frac{5}{2} \]

The parabola y = -2x² + 7x - 5

xy
-1.25-16.875
-0.25-6.875
0.75-0.875
1.751.125
2.75-0.875
3.75-6.875
4.75-16.875

Questions

What are the solutions of -2x² + 7x - 5 = 0?

The solutions of -2x² + 7x - 5 = 0 are x = 1 or x = 5/2.

Does -2x² + 7x - 5 = 0 factor?

Yes: -1(x-1)(2x-5) = 0.

What is the vertex of y = -2x² + 7x - 5?

(7/4, 9/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 7/2, product = c/a = 5/2.

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