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Solve 5x² + 7x - 6 = 0

The solutions of 5x² + 7x - 6 = 0 are x = 3/5 or x = -2.

x² +x += 0
5x2+7x−6=0⇒x=35, −25x^2 + 7x - 6 = 0 \Rightarrow x = \frac{3}{5},\ -2
Discriminant D = 169 · x ≈ 0.6, -2

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

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Graph of y = 5x² + 7x - 6-4-3-2-1123-10102030405060x = 0.6x = -2vertexy = 5x² + 7x - 6
Graph of y = 5x² + 7x - 6. The curve crosses the x-axis at x = 0.6 and x = -2; the vertex is at (-0.7, -8.45).

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 = -6.

    5x2+7x−6=05x^2 + 7x - 6 = 0
  2. 2

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

    D=72−4⋅5⋅(−6)=49+120=169D = 7^2 - 4\cdot5\cdot(-6) = 49 + 120 = 169
  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−3)(x+2)=0\left(5x - 3\right)\left(x + 2\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−3=0⇒x=35x+2=0⇒x=−25x - 3 = 0 \Rightarrow x = \frac{3}{5}\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−7±16910x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{169}}{10}
  7. 7

    Simplify to get the solutions:

    x=35orx=−2x = \frac{3}{5}\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = 3/5 back in: a·x² + b·x + c = 5·(9/25) + 7·(3/5) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 7x - 6 = 0\ \Rightarrow\ x = \frac{3}{5},\ -2 \]
\[ 5x^2 + 7x - 6 = 0 \]
\[ D = 7^2 - 4\cdot5\cdot(-6) = 49 + 120 = 169 \]
\[ \left(5x - 3\right)\left(x + 2\right) = 0 \]
\[ 5x - 3 = 0 \Rightarrow x = \frac{3}{5}\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{169}}{10} \]
\[ x = \frac{3}{5}\quad\text{or}\quad x = -2 \]

The parabola y = 5x² + 7x - 6

xy
-3.736.55
-2.711.55
-1.7-3.45
-0.7-8.45
0.3-3.45
1.311.55
2.336.55

Questions

What are the solutions of 5x² + 7x - 6 = 0?

The solutions of 5x² + 7x - 6 = 0 are x = 3/5 or x = -2.

Does 5x² + 7x - 6 = 0 factor?

Yes: (5x-3)(x+2) = 0.

What is the vertex of y = 5x² + 7x - 6?

(-7/10, -169/20). 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 = -6/5.

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