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Solve 5x² - 2x - 7 = 0

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

x² +x += 0
5x2−2x−7=0⇒x=75, −15x^2 - 2x - 7 = 0 \Rightarrow x = \frac{7}{5},\ -1
Discriminant D = 144 · x ≈ 1.4, -1

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

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Graph of y = 5x² - 2x - 7-3-2-1123-10102030405060x = 1.4x = -1vertexy = 5x² - 2x - 7
Graph of y = 5x² - 2x - 7. The curve crosses the x-axis at x = 1.4 and x = -1; the vertex is at (0.2, -7.2).

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 = -2, c = -7.

    5x2−2x−7=05x^2 - 2x - 7 = 0
  2. 2

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

    D=(−2)2−4⋅5⋅(−7)=4+140=144D = (-2)^2 - 4\cdot5\cdot(-7) = 4 + 140 = 144
  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:

    (5x−7)(x+1)=0\left(5x - 7\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:

    5x−7=0⇒x=75x+1=0⇒x=−15x - 7 = 0 \Rightarrow x = \frac{7}{5}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=2±14410x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{144}}{10}
  7. 7

    Simplify to get the solutions:

    x=75orx=−1x = \frac{7}{5}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 7/5 back in: a·x² + b·x + c = 5·(49/25) + (-2)·(7/5) + (-7) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 2x - 7 = 0\ \Rightarrow\ x = \frac{7}{5},\ -1 \]
\[ 5x^2 - 2x - 7 = 0 \]
\[ D = (-2)^2 - 4\cdot5\cdot(-7) = 4 + 140 = 144 \]
\[ \left(5x - 7\right)\left(x + 1\right) = 0 \]
\[ 5x - 7 = 0 \Rightarrow x = \frac{7}{5}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{144}}{10} \]
\[ x = \frac{7}{5}\quad\text{or}\quad x = -1 \]

The parabola y = 5x² - 2x - 7

xy
-2.837.8
-1.812.8
-0.8-2.2
0.2-7.2
1.2-2.2
2.212.8
3.237.8

Questions

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

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

Does 5x² - 2x - 7 = 0 factor?

Yes: (5x-7)(x+1) = 0.

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

(1/5, -36/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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