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Solve 5x² + 7x + 2 = 0

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

x² +x += 0
5x2+7x+2=0⇒x=−25, −15x^2 + 7x + 2 = 0 \Rightarrow x = -\frac{2}{5},\ -1
Discriminant D = 9 · x ≈ -0.4, -1

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

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Graph of y = 5x² + 7x + 2-3-2-11210203040x = -0.4x = -1vertexy = 5x² + 7x + 2
Graph of y = 5x² + 7x + 2. The curve crosses the x-axis at x = -0.4 and x = -1; the vertex is at (-0.7, -0.45).

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 = 7, c = 2.

    5x2+7x+2=05x^2 + 7x + 2 = 0
  2. 2

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

    D=72−4⋅5⋅2=49−40=9D = 7^2 - 4\cdot5\cdot2 = 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:

    (5x+2)(x+1)=0\left(5x + 2\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+2=0⇒x=−25x+1=0⇒x=−15x + 2 = 0 \Rightarrow x = -\frac{2}{5}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−7±910x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{9}}{10}
  7. 7

    Simplify to get the solutions:

    x=−25orx=−1x = -\frac{2}{5}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = -2/5 back in: a·x² + b·x + c = 5·(4/25) + 7·(-2/5) + 2 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 7x + 2 = 0\ \Rightarrow\ x = -\frac{2}{5},\ -1 \]
\[ 5x^2 + 7x + 2 = 0 \]
\[ D = 7^2 - 4\cdot5\cdot2 = 49 - 40 = 9 \]
\[ \left(5x + 2\right)\left(x + 1\right) = 0 \]
\[ 5x + 2 = 0 \Rightarrow x = -\frac{2}{5}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{9}}{10} \]
\[ x = -\frac{2}{5}\quad\text{or}\quad x = -1 \]

The parabola y = 5x² + 7x + 2

xy
-3.744.55
-2.719.55
-1.74.55
-0.7-0.45
0.34.55
1.319.55
2.344.55

Questions

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

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

Does 5x² + 7x + 2 = 0 factor?

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

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

(-7/10, -9/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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