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Solve 5x² + 12x + 7 = 0

The solutions of 5x² + 12x + 7 = 0 are x = -1 or x = -7/5.

x² +x += 0
5x2+12x+7=0⇒x=−1, −755x^2 + 12x + 7 = 0 \Rightarrow x = -1,\ -\frac{7}{5}
Discriminant D = 4 · x ≈ -1, -1.4

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

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Graph of y = 5x² + 12x + 7-3-2-11-55101520253035x = -1x = -1.4vertexy = 5x² + 12x + 7
Graph of y = 5x² + 12x + 7. The curve crosses the x-axis at x = -1 and x = -1.4; the vertex is at (-1.2, -0.2).

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 = 12, c = 7.

    5x2+12x+7=05x^2 + 12x + 7 = 0
  2. 2

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

    D=122−4⋅5⋅7=144−140=4D = 12^2 - 4\cdot5\cdot7 = 144 - 140 = 4
  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)(5x+7)=0\left(x + 1\right)\left(5x + 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=−15x+7=0⇒x=−75x + 1 = 0 \Rightarrow x = -1\qquad 5x + 7 = 0 \Rightarrow x = -\frac{7}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−12±410x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{4}}{10}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -1 back in: a·x² + b·x + c = 5·(1) + 12·(-1) + 7 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 12x + 7 = 0\ \Rightarrow\ x = -1,\ -\frac{7}{5} \]
\[ 5x^2 + 12x + 7 = 0 \]
\[ D = 12^2 - 4\cdot5\cdot7 = 144 - 140 = 4 \]
\[ \left(x + 1\right)\left(5x + 7\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 5x + 7 = 0 \Rightarrow x = -\frac{7}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{4}}{10} \]
\[ x = -1\quad\text{or}\quad x = -\frac{7}{5} \]

The parabola y = 5x² + 12x + 7

xy
-4.244.8
-3.219.8
-2.24.8
-1.2-0.2
-0.24.8
0.819.8
1.844.8

Questions

What are the solutions of 5x² + 12x + 7 = 0?

The solutions of 5x² + 12x + 7 = 0 are x = -1 or x = -7/5.

Does 5x² + 12x + 7 = 0 factor?

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

What is the vertex of y = 5x² + 12x + 7?

(-6/5, -1/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -12/5, product = c/a = 7/5.

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