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Solve 4x² - 10x - 6 = 0

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

x² +x += 0
4x2−10x−6=0⇒x=3, −124x^2 - 10x - 6 = 0 \Rightarrow x = 3,\ -\frac{1}{2}
Discriminant D = 196 · x ≈ 3, -0.5

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

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Graph of y = 4x² - 10x - 6-3-2-112345-20-10102030405060x = 3x = -0.5vertexy = 4x² - 10x - 6
Graph of y = 4x² - 10x - 6. The curve crosses the x-axis at x = 3 and x = -0.5; 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=(−10)2−4⋅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(x−3)(2x+1)=02\left(x - 3\right)\left(2x + 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:

    x−3=0⇒x=32x+1=0⇒x=−12x - 3 = 0 \Rightarrow x = 3\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}
  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=3orx=−12x = 3\quad\text{or}\quad x = -\frac{1}{2}
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 4·(9) + (-10)·(3) + (-6) = 0 ✓

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

The parabola y = 4x² - 10x - 6

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

Questions

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

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

Does 4x² - 10x - 6 = 0 factor?

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