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Solve 4x² + 10x - 6 = 0

The solutions of 4x² + 10x - 6 = 0 are x = 1/2 or x = -3.

x² +x += 0
4x2+10x−6=0⇒x=12, −34x^2 + 10x - 6 = 0 \Rightarrow x = \frac{1}{2},\ -3
Discriminant D = 196 · x ≈ 0.5, -3

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

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Graph of y = 4x² + 10x - 6-5-4-3-2-1123-20-10102030405060x = 0.5x = -3vertexy = 4x² + 10x - 6
Graph of y = 4x² + 10x - 6. The curve crosses the x-axis at x = 0.5 and x = -3; the vertex is at (-1.25, -12.25).

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 = 10, c = -6.

    4x2+10x−6=04x^2 + 10x - 6 = 0
  2. 2

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

    D=102−4⋅4⋅(−6)=100+96=196D = 10^2 - 4\cdot4\cdot(-6) = 100 + 96 = 196
  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:

    2(2x−1)(x+3)=02\left(2x - 1\right)\left(x + 3\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    2x−1=0⇒x=12x+3=0⇒x=−32x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−10±1968x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{196}}{8}
  7. 7

    Simplify to get the solutions:

    x=12orx=−3x = \frac{1}{2}\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = 1/2 back in: a·x² + b·x + c = 4·(1/4) + 10·(1/2) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 + 10x - 6 = 0\ \Rightarrow\ x = \frac{1}{2},\ -3 \]
\[ 4x^2 + 10x - 6 = 0 \]
\[ D = 10^2 - 4\cdot4\cdot(-6) = 100 + 96 = 196 \]
\[ 2\left(2x - 1\right)\left(x + 3\right) = 0 \]
\[ 2x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{196}}{8} \]
\[ x = \frac{1}{2}\quad\text{or}\quad x = -3 \]

The parabola y = 4x² + 10x - 6

xy
-4.2523.75
-3.253.75
-2.25-8.25
-1.25-12.25
-0.25-8.25
0.753.75
1.7523.75

Questions

What are the solutions of 4x² + 10x - 6 = 0?

The solutions of 4x² + 10x - 6 = 0 are x = 1/2 or x = -3.

Does 4x² + 10x - 6 = 0 factor?

Yes: 2(2x-1)(x+3) = 0.

What is the vertex of y = 4x² + 10x - 6?

(-5/4, -49/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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