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Solve 4x² + 12x - 10 = 0

The solutions of 4x² + 12x - 10 = 0 are x = (-3 + √19)/2 or x = (-3 - √19)/2, that is x ≈ 0.67944947 and x ≈ -3.6794495.

x² +x += 0
4x2+12x−10=0⇒x=−3+192, −3−1924x^2 + 12x - 10 = 0 \Rightarrow x = \frac{-3 + \sqrt{19}}{2},\ \frac{-3 - \sqrt{19}}{2}
Discriminant D = 304 · x ≈ 0.679449, -3.67945

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

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Graph of y = 4x² + 12x - 10-6-5-4-3-2-1123-20204060x = 0.67944947x = -3.6794495vertexy = 4x² + 12x - 10
Graph of y = 4x² + 12x - 10. The curve crosses the x-axis at x = 0.67944947 and x = -3.6794495; the vertex is at (-1.5, -19).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = 12, c = -10.

    4x2+12x−10=04x^2 + 12x - 10 = 0
  2. 2

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

    D=122−4⋅4⋅(−10)=144+160=304D = 12^2 - 4\cdot4\cdot(-10) = 144 + 160 = 304
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−12±3048x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{304}}{8}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=−3+192orx=−3−192x = \frac{-3 + \sqrt{19}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{19}}{2}
  6. 6

    As decimals: x ≈ 0.67944947 and x ≈ -3.6794495.

  7. 7

    Check numerically: for x ≈ 0.67944947 the left side is 3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 4x^2 + 12x - 10 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{19}}{2},\ \frac{-3 - \sqrt{19}}{2} \]
\[ 4x^2 + 12x - 10 = 0 \]
\[ D = 12^2 - 4\cdot4\cdot(-10) = 144 + 160 = 304 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{304}}{8} \]
\[ x = \frac{-3 + \sqrt{19}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{19}}{2} \]

The parabola y = 4x² + 12x - 10

xy
-4.517
-3.5-3
-2.5-15
-1.5-19
-0.5-15
0.5-3
1.517

Questions

What are the solutions of 4x² + 12x - 10 = 0?

The solutions of 4x² + 12x - 10 = 0 are x = (-3 + √19)/2 or x = (-3 - √19)/2, that is x ≈ 0.67944947 and x ≈ -3.6794495.

Does 4x² + 12x - 10 = 0 factor?

Not over the integers, because the discriminant 304 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 4x² + 12x - 10?

(-3/2, -19). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = -5/2.

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    Solve 4x² + 12x - 10 = 0: x = (-3 + √19)/2 or x = (-3 - √19)/2 (steps) | WordGPT