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Solve 4x² - 10x + 6 = 0

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

x² +x += 0
4x2−10x+6=0⇒x=32, 14x^2 - 10x + 6 = 0 \Rightarrow x = \frac{3}{2},\ 1
Discriminant D = 4 · x ≈ 1.5, 1

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

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Graph of y = 4x² - 10x + 6-1123451015202530x = 1.5x = 1vertexy = 4x² - 10x + 6
Graph of y = 4x² - 10x + 6. The curve crosses the x-axis at x = 1.5 and x = 1; the vertex is at (1.25, -0.25).

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=4D = (-10)^2 - 4\cdot4\cdot6 = 100 - 96 = 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:

    2(2x−3)(x−1)=02\left(2x - 3\right)\left(x - 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:

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

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=10±48x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{4}}{8}
  7. 7

    Simplify to get the solutions:

    x=32orx=1x = \frac{3}{2}\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = 3/2 back in: a·x² + b·x + c = 4·(9/4) + (-10)·(3/2) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 10x + 6 = 0\ \Rightarrow\ x = \frac{3}{2},\ 1 \]
\[ 4x^2 - 10x + 6 = 0 \]
\[ D = (-10)^2 - 4\cdot4\cdot6 = 100 - 96 = 4 \]
\[ 2\left(2x - 3\right)\left(x - 1\right) = 0 \]
\[ 2x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{4}}{8} \]
\[ x = \frac{3}{2}\quad\text{or}\quad x = 1 \]

The parabola y = 4x² - 10x + 6

xy
-1.7535.75
-0.7515.75
0.253.75
1.25-0.25
2.253.75
3.2515.75
4.2535.75

Questions

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

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

Does 4x² - 10x + 6 = 0 factor?

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

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

(5/4, -1/4). 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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